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Feynman’s OperationalCalculus and BeyondNoncommutativity and Time-Ordering

Gerald W. JohnsonMichel L. Lapidusand

Lance Nielsen

3

Feynman’s Operational Calculus and Beyond. First Edition. Gerald W. Johnson, Michel L. Lapidus and Lance Nielsen.© Gerald W. Johnson, Michel L. Lapidus and Lance Nielsen 2015. Published in 2015 by Oxford University Press.

3

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DEDICATIONS

To Gerald Johnson (Jerry), my wonderful coauthor, long-time friend and collaborator, for allhis unique qualities, including his humor, incredible integrity, loyalty, generosity and

perspicacity.

To his wife, Joan, his life-long companion and best friend, with gratitude for her friendshipand for all she does for Jerry.

To my own wife and companion, Odile, the love of my life, without whom I cannot live.

To the memory of Richard Feynman andMark Kac, friends and colleagues, with whom weshared many thoughts and joyful moments.

Michel L. Lapidus

I first met Jerry Johnson after asking some faculty at the University of Nebraska, Lincoln, ifthere was anyone in the department of mathematics who worked with path integrals. I was toldto talk to Dr. Johnson, who was finishing class in such-and-such a room. I found this room and,while Jerry was erasing the chalk board, I asked if he knew of a good book concerning pathintegrals. The response was, with a smile, “I’m writing one.” Of course, the book in question isthat written by Jerry and Michel, i.e., The Feynman Integral and Feynman’s OperationalCalculus [114]. This happenstance, some two decades ago, started me down the path that I

continue to follow today.

As time moved on, Jerry consented to be my advisor, and I continue to be very thankful that Iwas lucky enough to have had someone with Jerry’s wisdom, integrity, humor and patience as amentor and friend during my time as a Ph.D. candidate and during my years in academia. It isbecause of Jerry that I learned how to do and write mathematics and, were it not for him, I

would not be a coauthor of this volume. It has been a singular privilege to work on this volumewith Jerry and also with Michel, both mathematicians of enormous talent and individuals I hold

in the highest esteem.

Finally, I’d like to thank my dearest friend Amy for her good-humored tolerance to mycontinual protestations that “I can’t, I have to work on the book.” Amy and her youngest sonSam (as well as the family dog Lucky!) often gave me a needed escape from the manuscript

when I could not bear to look at LATEX any longer.

Lance Nielsen

ACKNOWLEDGMENTS

Michel Lapidus’s research described in parts of this book was partially supported by theU.S. National Science Foundation (NSF) under grants DMS-0707524 and DMS-1107750(as well as by many earlier NSF research grants since the mid-1980s).

The second author would also like to express his gratitude to many research insti-tutes at which he was a visiting professor while aspects of this research program weredeveloped, including especially the Mathematical Sciences Research Institute (MSRI) inBerkeley, California, USA, the Erwin Schrödinger International Institute of MathematicalPhysics in Vienna, Austria, and the Institut des Hautes Etudes Scientifiques (IHES) inBures-sur-Yvette, Paris, France.

Michel Lapidus

I would like to express my gratitude to the attendees—Jerry Johnson and Dave Skougamong others—of the functional integration seminar at the University of Nebraska,Lincoln, for the many opportunities to present and discuss my research, parts of whichappear in this book. I would also like to thank the second author, Michel Lapidus, for pro-viding a portion of the funding for a trip to Riverside, California, in 2009, to begin theprocess of writing this volume.

Lance Nielsen

PREFACE

This book is aimed at providing a coherent, essentially self-contained, rigorous and com-prehensive abstract theory of Feynman’s operational calculus for noncommuting operators.Although it is inspired by Feynman’s original heuristic suggestions and time-ordering rulesin his seminal paper [58], as will be made abundantly clear in the introduction (Chapter 1)and elsewhere in the text, the theory developed in this book also goes well beyond them ina number of directions which were not anticipated in Feynman’s work. Hence, the secondpart of the main title of this book.

It may be helpful to the reader to situate the present research monograph relative toa companion book [114], written by the first two named authors (Gerald Johnson andMichel Lapidus) and titled The Feynman Integral and Feynman’s Operational Calculus. (Letus reassure the reader at once that [114] is not a prerequisite for the present book, however,as will be discussed in more detail further on in this preface.) The latter nearly 800-pagebook [114] was initially published in 2000 by Oxford University Press (with a paperbackedition in 2002 and an electronic edition in the late 2000s) in the same series as the presentmonograph. It provides a number of different approaches to the Feynman path integral (or“sums over histories”), in both “real” and “imaginary” time.

Beginning with Chapter 14 and ending with Chapter 18, the second part of [114] (based,in part, on [110–113] along with [137–143]) develops a rigorous theory of Feynman’soperational calculus, using certain operator-valued Wiener and Feynman path integrals(called “analytic-in-mass Feynman integrals”) as well as associated commutative Banachalgebras of functionals, called “disentangling algebras,” and corresponding noncommutat-ive operations (namely, a noncommutative addition and multiplication) acting on them.The resulting time-indexed family of disentangling algebras, along with the associated non-commutative operations, provides a rich algebraic, analytic and combinatorial structure forthe development of a concrete theory of Feynman’s operational calculus within the contextof Feynman path integrals and related path or stochastic integrals.

On the other hand, Chapter 19 of [114] (based on the earlier joint work of the authors of[114] with Brian DeFacio in [33, 34]) begins to build a bridge between the above rigorousconcrete version of the operational calculus and a possible, more general operational calcu-lus valid for abstract operators (acting on Banach or Hilbert spaces) not necessarily arisingvia Wiener or Feynman functionals and associated path integrals. The connections with alarge class of associated evolution equations are also studied in Chapter 19 of [114].

In a sense, Chapters 15–18 together with, specifically, Chapter 19 of [114] lay the foun-dations and provide a possible starting point for the development of a fully rigorous andmore abstract theory of Feynman’s operational calculus, which is the object of the presentbook. The reader familiar with Chapters 15–19 of [114] will recognize some aspects of,and motivations for, the theory developed in the present book, but in essence (with the

x | preface

notable exception of Chapter 19 of [114], which inherently serves as the basis for much ofChapter 6 of this monograph and is described in part in Section 6.2), the two theories andtheir presentations are essentially distinct and independent of one another. In particular, thepresent theory is aimed at dealing with abstract (typically) noncommuting operators, ratherthan operators arising from some kind of path integration (viewed as a suitable quantizationprocedure), as in [114]. In fact, some of the key structures developed in the present book(particularly, the family of commutative disentangling algebras, the corresponding disentan-gling maps and the associated noncommutative operations; see Chapters 2, 5 and 6) enableus, in some sense, to obtain an appropriate abstract substitute for a generalized functionalintegral (viewed as a suitable “quantization procedure” (in the sense of [143] and as de-scribed in [114, Section 18.6]) associated with the Feynman operational calculus attachedto a given n-tuple of pairs

{(Aj,μj

)}nj=1 of typically noncommuting bounded operators Aj

and probability measuresμj, for j = 1, . . . , n and n ≥ 2).As mentioned earlier, the present book is essentially self-contained. In particular, the

earlier book [114] is not a prerequisite for understanding its contents. However, the inter-ested reader may wish to consult Chapters 7 and 14 of [114], which provide a thoroughintroduction to the physical and heuristic aspects of “the” Feynman integral and Feynman’soperational calculus, respectively, as well as to the associated and rather daunting math-ematical difficulties. In the present book, we assume only that the reader has a reasonablegraduate-level background in analysis, measure theory and functional analysis or operatortheory.1 Much of the necessary remaining background material is provided in the text itself.

In the introduction (Chapter 1) of this research monograph and elsewhere in the rest ofthe text (for example, in parts of Chapters 2, 3, 5, 6 and 8), we will present an overview ofthe heuristic and physical aspects of Feynman’s operational calculus, with an eye towardsthe rigorous abstract theory developed in the book, based on time-ordering, noncommut-ativity, disentangling algebras, and associated disentangling maps and noncommutativeoperations. All of these notions will be progressively introduced and precisely defined, be-ginning with Chapter 2 and continuing on to Chapter 6, in particular. Along the way, severaltechniques for carrying out the “disentangling process,” which is at the heart of Feynman’sheuristic operator calculus proposed in [58], are developed throughout the book. See,for example, the discussion of the “disentangling of an exponential factor” (in Section 3.4and, much more generally, in Chapter 6), the extraction of multilinear factors and itera-tive disentangling (in Chapter 4), the disentangling formulas (obtained in Chapter 5),the generalized Dyson expansions along with the corresponding evolution equations (inChapter 6), the discussion of disentangling via the use of continuous and discrete meas-ures (in Chapter 8), and the “derivation formulas” (via suitable functional derivatives inChapter 9).

Reflecting upon the contents of this book, one sees in hindsight that the variety of disen-tangling techniques developed in the present theory constitutes one of its main features and

1 See, for example, [13, 26, 41, 187, 192] for textbooks on these basic subjects; see also [11, 44, 78, 83, 123,124, 188, 193, 195, 214] along with [114, Chapters 3, 6–10, 12 and 15] for more advanced material which willoccasionally be needed in this book.

preface | xi

lies at the core of the present theory. We hope that the reader will find these disentanglingresults useful for his or her own purposes and will be stimulated to enrich the theory withnew results, techniques and perspectives of an analytical, geometric, combinatorial or alge-braic nature. The epilogue to this book (Chapter 11) has been written so as to facilitate thisprocess and to suggest several possible directions for future research extending Feynman’soperational calculus in a variety of ways.

Gerald W. Johnson, Michel L. Lapidus and Lance NielsenMarch 2015

CONTENTS

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Disentangling: Definitions, Properties and ElementaryExamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.1 The Disentangling Algebras 352.2 The Disentangling Maps 392.3 Simple Examples of Disentangling 492.4 The Effects of Commutativity 57

3 Disentangling via Tensor Products andOrdered Supports . . . . . . . . . 633.1 Disentangling via Direct Sums and Tensor Products 643.2 Nonprobability Measures 703.3 Disentangling via Measures with Ordered Supports 743.4 Disentangling an Exponential Factor 84

4 Extraction ofMultilinear Factors and Iterative Disentangling . . . . . . . 954.1 Extraction of Linear Factors 974.2 Extraction of Bilinear Factors 1084.3 Extraction of Multilinear Factors 1124.4 Iterated, but not Multilinear, Disentangling 1154.5 Consequences and Examples 1184.6 Appendix: Decomposing Disentangling 121

5 Auxiliary Operations andDisentangling Algebras. . . . . . . . . . . . . . . .1375.1 The Noncommutative Operations⊗ and

◦+ 138

5.2 Additional Examples of Disentangling Formulas 1515.3 Relationship to the Disentangling Maps 152

6 Time-Dependent Feynman’s Operational Calculus andEvolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .157

6.1 Initial Definitions 1606.2 The Time Evolution of the Disentangled Exponential Function 1656.3 The Rigorous Definition of the Disentangling Map

in the Presence of an Unbounded Operator: Exponential Factors 2016.4 A Generalized Integral Equation for Feynman’s Operational Calculus 212

7 Stability Properties of Feynman’s Operational Calculi . . . . . . . . . . . .2297.1 The General Setting for the Stability Theory 2357.2 Joint Stability: Stability with Respect to the Operators and the

Time-Ordering Measures 246

xiv | contents

7.3 Stability with Respect to the Operators 2537.4 Stability with Respect to the Time-Ordering Measures 256

8 Disentangling via Continuous andDiscreteMeasures. . . . . . . . . . . . .2618.1 Definitions and Notation 2628.2 Time-Ordering Monomials 2668.3 Definition of the Disentangling Map 2698.4 Examples 2778.5 Stability Results 286

9 Derivational Derivatives and Feynman’s Operational Calculi . . . . . . . 3139.1 Introduction 3139.2 Disentangling Maps, Homomorphisms and Antihomomorphisms 3149.3 The Derivation Formula 3189.4 Higher-Order Expansions 323

10 Spectral Theory for Noncommuting Operators . . . . . . . . . . . . . . . . .32710.1 Introduction 32710.2 Background Material on Distributions 32810.3 Functional Calculus for Noncommuting Operators 331

11 Epilogue:Miscellaneous Topics and Possible Extensions . . . . . . . . . .34311.1 Overview 34311.2 Open Problems and Future Research Directions 344

References 355Notation Index 365Subject Index 367

1

Introduction

The ideas developed in this volume are rooted in the works of Richard Feynman in the mid-twentieth century. Before we provide an overview of this monograph, it seems worthwhileto look at the historical roots of Feynman’s ideas as they relate to his operator calculus andthe closely related path integrals. Indeed, while many papers concerned with Feynman’soperational calculus cite Feynman’s 1951 paper [58] as the starting point,1 the roots ofthe operational calculus go back to Feynman’s undergraduate days at MIT. As he states inhis Nobel lecture “The development of the space–time view of quantum electrodynamics”(December 11, 1965, reprinted in [15]; all quotes from Feynman’s Nobel lecture comefrom the transcript contained in [15]):

I worked on this problem about eight years until the final publication in 1947. The be-ginning of the thing was at the Massachusetts Institute of Technology [MIT], whenI was an undergraduate student reading about the known physics, learning slowlyabout all these things that people were worrying about, and realizing ultimately thatthe fundamental problem of the day was that the quantum theory of electricity andmagnetism was not completely satisfactory. This I gathered from books like those ofHeitler and Dirac. I was inspired by the remarks in these books; not by the parts inwhich everything was proved and demonstrated carefully and calculated, because Icouldn’t understand those very well. At [that] young age, what I could understandwere the remarks about the fact that this doesn’t make any sense, and the last sentenceof the book of Dirac I can still remember, “It seems that some essentially new physicalideas are here needed”. So, I had this as a challenge and an inspiration. I also had a per-sonal feeling, that since they didn’t get a satisfactory answer to the problem I wantedto solve, I don’t have to pay a lot of attention to what they did do.

(It is interesting to note that, during the summer following Feynman’s sophomore yearat MIT, he tried to invent an operator calculus by attempting to develop rules for the differ-entiation and integration of noncommuting variables. He wrote: “Now I think I’m wrongon account of those darn partial integrations. I oscillate between right and wrong.” Further,

1 A number of references (from the mathematics or physics literature) on, or related to, Feynman’s operationalcalculus can be found at the very end of this introduction, as well as in the epilogue (Chapter 11).

Feynman’s Operational Calculus and Beyond. First Edition. Gerald W. Johnson, Michel L. Lapidus and Lance Nielsen.© Gerald W. Johnson, Michel L. Lapidus and Lance Nielsen 2015. Published in 2015 by Oxford University Press.

2 | introduction

“Hot dog! after 3 wks of work. . . I have at last found a simple proof. It’s not importantto write it, however. The only reason I wanted to do it was because I couldn’t do it andfelt that there were some more relations between the An & their derivatives that I had notdiscovered. . . Maybe I’ll get electricity into the metric yet!” ([76, p. 75].)

During his Nobel lecture, Feynman proceeds to summarize his ultimately mistaken ideaswhich he had at MIT concerning electron self-interaction, as well as the presence of an in-finite number of degrees of freedom in the electromagnetic field. Nevertheless, the ideasFeynman had as an undergraduate student at MIT had a tremendous influence on the de-velopment of his space–time theory of quantum electrodynamics. As he says, somewhatfurther on in his Nobel lecture:

and the idea seemed so obvious to me and so elegant that I fell deeply in love withit. And, like falling in love with a woman, it is only possible if you do not know muchabout her, so you cannot see her faults. The faults will become apparent later, but afterthe love is strong enough to hold you to her. So, I was held to this theory, in spite of alldifficulties, by my youthful enthusiasm.

Feynman was a graduate student at Princeton University, working with John Wheeleron an action-at-a-distance theory of classical electrodynamics, when his early ideas beganto bear fruit. He and Wheeler found that they could reformulate their work via a principleof least action. They were able to find a form for an action that involved the motions of thecharges only, which upon variation would give the equations of motions of the charges. (Infact, the action was

A =∑i

mi

∫ (XiμX

iμ

)1/2 dαi +12

∑i�=j

eiej∫ ∫

δ(I2ij

)Xiμ(αi)Xj

μ(αj) dαi dαj,

where the indices i and j label the interacting charged particles involved and, for i �= j,

I2ij =

[Xiμ(αi) – Xj

μ(αj)] [Xiμ(αi) – Xj

μ(αj)]

is the square of the space–time distance between points on the paths. Here, mi andei are the mass and the electric charge, respectively of the i-th particle, and

{Xiμ

}and{

Xiμ

}denote the coordinates of its position and velocity, respectively.) From this action,

Feynman and Wheeler were able to obtain classical electrodynamics without appealing tothe electromagnetic field. Feynman remarks, in the Nobel lecture, that

I would also like to emphasize that by this time I was becoming used to a physical pointof view different from the more customary point of view. In the customary view, thingsare discussed as a function of time in very great detail. For example, you have the fieldat this moment, a differential equation gives you the field at the next moment and soon; a method, which I shall call the Hamilton method, the time differential method.We have, instead a thing that describes the character of the path throughout all ofspace and time. The behavior of nature is determined by saying her whole space–timepath has a certain character.

introduction | 3

Feynman had now solved the problem of classical electrodynamics in a way that wascompletely consistent with his original ideas while at MIT. All that was left was to makea quantum theory that was analogous to the classical theory. As is well known, if theaction for the classical theory is of the form of the integral of the Lagrangian of the vel-ocities and positions at the same time, then you can start with the Lagrangian and derivea Hamiltonian, which will then allow the quantum mechanics to be worked out. However,the action Feynman and Wheeler had (and that is given above) involves positions at twodifferent times, and therefore there was no clear way to develop the quantum mechanicalanalog. It was not until he met Herbert Jehle, who showed him Dirac’s paper in which theLagrangian comes into quantum mechanics, that Feynman was able to quantize his classicalelectrodynamics. Feynman says, in the Nobel lecture:

So, I thought I was finding out what Dirac meant, but, as a matter of fact, had madethe discovery that what Dirac thought was analogous, was, in fact, equal. I had then, atleast, the connection between the Lagrangian and quantum mechanics, but still withwave functions and infinitesimal times.

After this, Feynman was able to use his Lagrangian formalism to compute the wavefunction at a finite time by using factors of the form eiεL, where i :=

√–1, leading to his

representation of quantum mechanics in terms of an action. This, in turn, led to his ideaof the amplitude for a path; i.e., for each possible way a particle can travel between twopoints in space–time, there is an associated amplitude. This amplitude is eiS/h, where S is theaction.

With the path formulation of quantum mechanics, it became possible for Feynman todescribe photon interactions. However, when the action had a delay and involved morethan one time, he could no longer deal with a wave function. Nevertheless, with the pathformulation, Feynman developed a new idea. He found that, if a source emits a particle andif a detector is present to receive this particle, he could talk about the probability amplitudethat the source will emit and the detector will receive it. Furthermore, this could be donewithout specifying the instant at which the source emits or the detector receives the particleand without attempting to specify the state at any time in between. In other words, he couldfind the probability amplitude for the entire experiment. Thanks to his path formulation,he also obtained a theory of quantum electrodynamics. Many of these ideas are discussedin Feynman’s 1948 paper “A space–time approach to non-relativistic quantum mechanics”[54]. Indeed, the abstract of this paper states:

Non-relativistic quantum mechanics is formulated here in a different way. It is, how-ever, mathematically equivalent to the familiar formulation. In quantum mechanicsthe probability of an event which can happen in several different ways is the abso-lute square of a sum of complex contributions, one from each alternative way. Theprobability that a particle will be found to have a path x(t) lying somewhere withina region of space time is the square of a sum of contributions, one from each pathin the region. The contribution from a single path is postulated to be an exponentialwhose (imaginary) phase is the classical action (in units of h) for the path in question.The total contribution from all paths reaching x, t from the past is the wave function

4 | introduction

ψ(x, t). This is shown to satisfy Schrödinger’s equation. The relation to matrix andoperator algebra is discussed. Applications are indicated, in particular to eliminate thecoordinates of the field oscillators from the equations of quantum electrodynamics.

It is, of course, the second to last sentence of this abstract that is of most interest to us.We find, in Section 8 of [54], titled “Operator algebras,” the expressions that will becomeknown as path integrals. We read the following text in Section 9 (page 381 of [54]):

The operators corresponding to functions of xk+1 will appear to the left of the op-erators corresponding to functions of xk, i.e., the order of terms in a matrix operatorproduct corresponds to an order in time of the corresponding factors in a functional. Thus,if the functional can and is written in such a way that in each term, factors corres-ponding to later times appear to the left of factors corresponding to earlier terms, thecorresponding operator can immediately be written down if the order of the operatorsis kept the same as in the functional.

Here is the operator-ordering convention that will be one part of what we shall refer to, laterin this volume, as Feynman’s “rules.” The reader may very well guess that the time-orderingof operators will play a crucial role in this book. As to the importance Feynman gives to thetime-ordering of operators, we see later on that

It should be remarked that this rule must be especially carefully adhered to whenquantities involving velocities or higher derivatives are involved.

Hence, the heuristic “rules” that Feynman develops and which are used (in a rigorousmanner) throughout this monograph make their appearance in [54].

As one may expect, Feynman makes use in other places of the ideas that eventually ledto the “time-ordering rules” developed in [58]. For example, the ideas of time-ordering ofoperator products arise in [57]. On page 445 of the aforementioned paper, we read

where F is any function of the coordinate x1 at time t1, x2 at time t2 up to xk, tk, and, itis important to notice, we have assumed t′′ > t1 > t2 > · · · > tk > t′.

As we noted above, it is in the paper titled “An operator calculus having application toquantum electrodynamics” [58] that Feynman presents his method of forming functionsof noncommuting operators, or disentangling. The paper starts as follows:

In this paper we suggest an alteration in the mathematical notation for handling oper-ators. This new notation permits a considerable increase in the ease of manipulation ofcomplicated expressions involving operators. No results which are new are obtainedin this way, but it does permit one to relate various formulas of operator algebra inquantum mechanics in a simpler manner than is often available.

Appearing somewhat further on in this paper is a comment that has inspired a good deal ofresearch over the years and, perhaps, could be taken as a motivation for much of the contentof the current volume:

introduction | 5

The mathematics is not completely satisfactory. No attempt has been made tomaintain mathematical rigor. The excuse is not that it is expected that rigorous dem-onstrations can be easily supplied. Quite the contrary, it is believed that to put thepresent methods on a rigorous basis may be quite a difficult task, beyond the abilitiesof the author.2

It is in Section 1 of [58] that Feynman records the heuristic “rules” that form the basisfor this monograph. In the first section of the paper, we read the following text:

The order of operation of operators is conventionally represented by the position inwhich the operators are written on the paper. Thus, the productAB of two operatorsAandB is to be distinguished from the product in reverse orderBA. The algebra of oper-ators is noncommutative, so that all of the ordinary algebra, calculus, and analysis withordinary numbers becomes of small utility for operators. Thus, for a single operator,α, ordinary functions of this operator, such as A = expα, can be defined, for example,by power series. These functions obey the rules of ordinary analysis even though αis an operator. But if another operator β is introduced with which α does not com-mute, the question of functions of the two variables α, β is beset with commutationdifficulties and the simplest theorems of analysis are lost. . . .

We shall change the usual notation of the theory of operators and indicate the orderin which operators are to operate by a different device. We attach an index to the oper-ator with the rule that the operator with higher index operates later. Thus, BA may bewritten B1A0 or A0B1. The order no longer depends on the position on the paper, sothat all of the ordinary processes of analysis may be applied as though A0 and B1 werecommuting numbers. It is only at the end of the calculation, when the quantities areto be interpreted as operators, that the indices 0 and 1 are of importance if one wishesto reconvert an expression to the usual notation.

The first paragraph quoted above makes it clear that Feynman recognizes the problems withforming functions of noncommuting operators. As for the second paragraph quoted, we seeFeynman spell out his heuristic rules for computing functions of noncommuting operators:

(1) Attach indices to the operators involved, with the understanding that an operatorwith a higher index operates later.

(2) With indices attached, form (or compute) the function of the operators, treatingthe operators involved (with indices attached) as if they are commuting.

(3) Restore the conventional ordering of the operators via their indices.

After these heuristic rules have been applied to the function exp(α + β) on pages 109 and110 of this same paper [58], we read the following, just after Equation (5):

2 That this was so was repeatedly stressed by Richard Feynman to the second author (Michel Lapidus) duringseveral enjoyable and stimulating conversations in the early 1980s, in which he strongly encouraged him to developa rigorous theory of his operator (or operational) calculus.

6 | introduction

This process of rearranging the form of expressions involving operators ordered byindices so that they may be written in conventional form we shall call disentanglingthe operators. This process is not always easy to perform and, in fact, is the centralproblem of this operator calculus.

The quote above contains the first appearance of the term “disentangle”; and the disen-tangling operation (“rule” 3 above) is indeed, as the reader will come to appreciate, usuallythe most difficult part of any given problem in the operational calculus. However, “rule” 1is also worth commenting on at this time. We find, in the second to last paragraph of page110 of [58], the following remarks concerning when a given operator will act (or, where itwill operate), as well as making some other, general comments:

A word about notation: Inasmuch as in mathematics and physics there are alreadymany uses of the subscript notation, very often we shall write α(s) for αs. In a sense,α(s) is a function of s, namely, in the sense that although the operator αmay be defin-ite, its order of operation is not—so that the operator plus a prescription of where it isto operate, α(s), is a function of s. Furthermore, there will be many cases in which theoperator actually depends explicitly on the parameter of order. In this case we shouldhave strictly to write αs(s) but will omit the subscript when no ambiguity will resultfrom the change.

We may remark in a general sense about the mathematical character of our expres-sions. Given an expression such as

∫ 10 β(s) ds, we are not concerned with evaluating

the integral, for the quantity when separated from other factors with which it might bemultiplied is incompletely defined. Thus, although

∫ 10 βs ds standing alone is equiva-

lent simply to β , this is far from true when∫ 1

0 βs ds is multiplied by other expressionssuch as exp

∫ 10 αs ds. Thus, we must consider the complete expression as a complete

functional of the argument functionsα(s),β(s), etc. With each such functional we areendeavoring to associate an operator. The operator depends on the functional in acomplex way (the operator is a functional of a functional) so that, for example, the op-erator corresponding to the product of two functionals is not (in general) the simpleproduct of the operators corresponding to the separate factors. (The correspondingstatement equating the sum of two functionals and the sum of the correspondingoperators is true, however.) Hence, we can consider the most complex expressionsinvolving a number of operatorsM,N, as described by functionals F

[M(s),N(s) · · · ]

of the argument functions M(s), N(s) · · · (≡ Ms,Ns · · · ). For each functional we areto find the corresponding operator in some simple form; that is, we wish to disentanglethe functional. One fact we know is that any analytic rearrangement may be performedwhich leaves the value of the functional unchanged for arbitrary M(s), N(s)· · · con-sidered as ordinary numerical functions. Besides, there are a few special operationswhich we may perform on F

[M(s),N(s) · · · ], to disentangle the expressions, which

are valid only because the functional does represent an operator according to our rules.These special operations (such as extracting an exponential factor discussed in Sec. 3)are, of course, proper to the new calculus; and our powers of analysis in this field willincrease as we develop more of them.

introduction | 7

Early in the quote above, it can be seen that Feynman had devised a way of determiningwhen an operator will act in products. This device is that of attaching a time index to theoperator via the Lebesgue measure, i.e., writing

α =∫ 1

0α(s) ds,

where α(s) := α for all s ∈ [0, 1]. This idea can be generalized, following the work ofJohnson and Lapidus [110–113], Lapidus [138–142], and DeFacio, Johnson and Lapidus[33, 34], as well as in the later work of Jefferies and Johnson [96–99], to the use of a Borelprobability measureμ to attach indices to operators via

α =∫

[0,1]α(s)μ(ds),

where, just as above, α(s) := α for all s ∈ [0, 1]. Of course, we can do this for every oper-ator under consideration in a given problem. As Feynman indicates, while the individualintegrals are trivial, it is when we generate expressions involving products (or, more ac-curately, sums of products) that the complexity of the disentangling process shows itself.Furthermore, as remarked by Feynman, the complete (disentangled) expression dependson all of the operators α(s), β(s), etc., and this dependence is complex. The reader will seethroughout this volume that this is indeed the case, even in the case of functions of only twovariables.

It was at the Oldstone Conference (April 1949) that Feynman’s approach to quantumelectrodynamics gained preeminence, and at about this time, he published a set of papersthat would set the stage for a new era in modern physics. After his path integral paper[54] came “A relativistic cut-off for classical electrodynamics” [52], “Relativistic cut-off forquantum electrodynamics” [53], “The theory of positrons” [55], “Space–time approachto quantum electrodynamics” [56], “Mathematical formulation of the quantum theory ofelectromagnetic interaction” [57] and, most importantly for us in the context of this vol-ume, “An operator calculus having applications in quantum electrodynamics” [58]. (Seealso [76, pp. 271–272].)

As a conclusion to this historical sketch of the development of the operational calculusand the closely related ideas of path integrals, we use some of Feynman’s own words in aseries of interviews and conversations with Jagdish Mehra [156, pp. 325–327] that tookplace in Austin, Texas, during April 1970 and, much later, in Pasadena, California, duringJanuary 1988. Feynman had the following to say about the papers [57] and [58], which,as Mehra notes, were the final articles in the sequence of his groundbreaking papers onquantum electrodynamics:

I had invented a new mathematical method [the operator calculus] for dealing withoperators according to a parameter which, to this day, I feel is a great invention, andwhich nobody uses for anything; nobody pays any attention to it. Some day it will berecognized as an important invention. I still think it is something very important, justas important as I felt when I first wrote it.

8 | introduction

I had used it to formulate quantum electrodynamics. I invented it to do that. It wasin fact the mathematical formulation that I expressed at the Pocono conference—thatwas this crazy language. Dates don’t mean anything. It was published in 1951, but ithad all been invented by 1948. I called it the operator calculus.

I published it at that time because, after I had given the rules, and proved that theywere the same as the other things [of Schwinger and Tomonaga], it was important [toshow it formally]. Dyson had already given a proof. People don’t bother to read myproof because it’s too elaborate and funny, odd notations and path integrals, etc., butI had to do it in my own way for my own purposes. My paper on the “Mathematicalformulation of the quantum theory of electromagnetic interaction” was a rather un-necessary paper, because Dyson had done it in some way, and all I wanted to say washow I did it. But the other paper, on the operator calculus was not completely empty;I felt it was important. In the years since I had invented it I had accumulated a wholelot of debris. For instance, I had noticed certain ways of representing spin-0 particleswith path integrals, I had the operator calculus, and a whole lot of other things whichI did not know where to put. Most of it was, of course, the operator calculus, but inthe various appendices I included a whole variety of other things. With this paper Idisgorged myself of all the things I had thought about in the context of quantum elec-trodynamics; this was an entire backlog of valuable things. I still think that the centralitem, the operator calculus was an important invention.

With this paper I had completed the project on quantum electrodynamics. I didn’thave anything else remain that required publishing. In these two papers, I put every-thing that I had done and thought should be published on the subject. And that wasthe end of my published work on this field.

It is therefore apparent that Feynman considered his operator calculus as an important con-tribution, years after its initial development. In the six decades that followed the appearanceof Feynman’s operator calculus, mathematicians of various stripes have endeavored to de-velop this operator calculus in a mathematically rigorous way. The present monograph is amajor effort in this direction.

As a final remark, Mehra, on page 327 of [156], ends his discussion of Feynman’smathematical formulation of quantum electrodynamics with a description of Feynman’sattendance at the 1962 Solvay Conference. Mehra notes that “His preoccupation with theproblems of quantum electrodynamics had been over for quite some time, but he wouldcontinue to make use of the physical conceptions and mathematical techniques he hadpioneered in this field.”

We now turn to a discussion of the approach to Feynman’s operational calculus thatis taken in this monograph.3 In view of the historical sketch above, the reader will not

3 We refer the interested reader to the Preface for a brief discussion of the relationship between the abstracttheory of Feynman’s operational calculus developed in this book and the more concrete (as well as path-integral-based) approach developed—via Wiener and Feynman integrals, disentangling algebras (of Wiener functionals)and associated noncommutation operations—in the first two authors’ earlier book [114]; see, especially, [114,Chapters 14–19], based in part on [110–113], [137–143] and [34]. More information about this topic and itsrelationship to the present theory is also provided in various places in the present book.

introduction | 9

be surprised that we will take, as the starting point for our discussion, Feynman’s 1951paper “An operator calculus having applications in quantum electrodynamics” ([58]). Asdiscussed above, it was in this paper that Feynman outlined his heuristic rules for thecomputation (or formation) of functions of several noncommuting operators. From themathematician’s point of view, what is needed is to find a way to make these heuristicrules mathematically rigorous. It is worth noting that much of the earlier mathematicalwork done with the disentangling process (the operational calculus) was done heuristic-ally. Feynman’s rules were applied and the disentangled operator computed without muchattention to the presence of noncommutativity. Once the disentangling process was com-plete, “theorems” were then proven to show that the disentangled expression obtained hadthe necessary properties required by the problem under consideration. See, for example,[33, 34, 114], among others. The approach taken in this monograph (following that takenin the papers [96–99]) is to create a “commutative world” where the computations re-quired by Feynman’s heuristic rules can be done in a mathematically rigorous fashion.We will then map the result into the noncommutative setting of an operator algebra. Amuch more detailed discussion of this process will be carried out in the remainder of thischapter.

As is well known, the functional calculus for a single bounded linear operator is extremelyrich and well developed; the same is true of the functional calculus for a finite number ofbounded linear operators, as long as these operators form a commuting family. However,as soon as the assumption of commutativity is dropped, ambiguities arise, and the develop-ment of a functional calculus becomes much more difficult. Examples of such ambiguitiesare easy to find. Consider the function f (x, y) = x2y of the real (or complex) variables xand y. If we choose noncommuting linear operators A and B, the question arises as to howto define f (A,B). Since AB �= BA, a choice has to be made about how to form the productA2B: Do we use A2B, ABA, BA2, or some other expression or sum of expressions involvingtwo factors of A and one of B? As we will see, beginning with Chapter 2, the expressionone obtains for a function f (A,B) of two (or more) noncommuting operators is intimatelyrelated to the idea of time-ordering of the operators in operator products. As we have seenabove, the idea of time-ordering was a crucial ingredient of Feynman’s approach in the 1951paper [58].

Motivated by his work concerning path integration in nonrelativistic quantum mech-anics and in quantum electrodynamics, Feynman gave, in his 1951 paper [58], a heuristicformulation of an operational calculus for noncommuting operators. Here, we will notdiscuss rigorous mathematics, leaving that for later chapters, but will instead discuss theheuristic ideas of Feynman that have been touched on previously.

As will be seen, Feynman makes unconventional use of rules and formulas in his operatorcalculus. Indeed, Feynman writes [58, p. 124]:

The physicist is very familiar with such a situation and satisfied with it, especially sincehe is confident that he can tell if the answer is physically reasonable. But mathemat-icians may be completely repelled by the liberties taken here. The liberties are takennot because the mathematical problems are considered unimportant. On the contrary,this appendix is written to encourage the study of these forms from a mathematical

10 | introduction

standpoint. In the meantime, just as a poet often has license from the rules of grammarand pronunciation, we should like to ask for “physicists’ license” from the rules ofmathematics in order to express what we wish to say in as simple a manner as possible.

Feynman had a different way of tracking the order of noncommuting operators inproducts and this was, as we have mentioned, one of the keys to his operator calculus.

Feynman’s time-ordering convention

Feynman used time indices to specify the ordering of operators in products. It is understoodthat operators with earlier time indices act to the right, or earlier, than operators with latertime indices. As an example, if A and B are operators, we take

A(s1)B(s2) :=

⎧⎪⎨⎪⎩AB if s2 < s1,BA if s1 < s2,undefined if s1 = s2.

The necessity for evaluating operator products at the same time arises later, in Chapter 8;also, it is a key requirement in the earlier approach to Feynman’s operational calculusdiscussed in [114, Chapters 14–18] and based on [110–113] and [137–143], as wellas in the later works [96–99, 108, 109, 115, 117, 166, 167], for example. But, for nowand for the sake of simplicity, we will take the definition above as our time-orderingconvention.

Feynman’s heuristic rules

Some of the “rules,” loosely described, for the operational calculus are as follows:

(1) Attach time indices to the operators to specify the order of operators in products.(2) With time indices attached, form functions of these operators by treating them as

though they were commuting.(3) Finally, “disentangle” the resulting expressions; that is, restore the conventional order-

ing of the operators.

Of the disentangling process, Feynman states [58, p. 110], “The process is not alwayseasy to perform and, in fact, is the central problem of this operator calculus.” One shouldnote that Feynman did not attempt to supply rigorous proofs of his results. In fact, it is notalways clear how Feynman’s rules are to be applied, even heuristically.

The initial question that one ought to ask, when considering the rules above, is howtime indices can be attached to operators. Of course, one or more of the operators underconsideration may come with time indices naturally attached. For example, this happenswhen operators of multiplication by time-dependent potentials are present, and also inconnection with the Heisenberg representation in quantum mechanics. If an operator doesnot depend on time, as happens most frequently in quantum mechanics and in the math-ematical literature, we require a mechanism for the attachment of time indices. Given a

introduction | 11

time independent operator A, Feynman, almost without exception, used the Lebesguemeasure to attach time indices by writing

A =1t

∫ t

0A(s) ds,

where A(s) := A for 0 ≤ s ≤ t. While it seems artificial, this method of attaching time in-dices is extremely useful and is crucial to the approach to Feynman’s operational calculustaken in this monograph.

We now take the time to present in some detail several elementary examples which serveto illustrate Feynman’s rules. In these examples, X is a Banach space (or complete normedspace, with a norm denoted by ‖ · ‖X) and L(X) is the space of bounded linear operatorson X, a Banach space in its own right, equipped with its usual norm

‖A‖L(X) = sup {‖Ax‖X : x ∈ X, ‖x‖X ≤ 1}.

Example 1.0.1 Consider the function f (x, y) = xy. Let A,B ∈ L(X) and associateLebesgue measure � on [0, 1] to each operator; i.e., we attach time indices to bothoperators, using Lebesgue measure. To be clear, this means that we write

A =∫ 1

0A(s) ds and B =

∫ 1

0B(s) ds,

whereA(s) := A and B(s) := B for all 0 ≤ s ≤ 1. We calculate f (A,B) as follows, nam-ing the result f�,�(A,B) in order to stress that Lebesgue measure has been used to attachtime indices to both A and B:

f�,�(A,B) ={∫ 1

0A(s) ds

}{∫ 1

0B(s) ds

}

=∫ 1

0

∫ 1

0A(s1)B(s2) ds1ds2

=∫

{(s1,s2):s1<s2}

B(s2)A(s1) ds1ds2 +∫

{(s1,s2):s2<s1}

A(s1)B(s2) ds1ds2

=12BA +

12AB

=12

(AB + BA) .

The first equality above follows from the fact that Lebesgue measure is a probabilitymeasure on [0, 1]. The second equality follows by writing the product of the two in-tegrals in the first line as a double integral of the function A(s1)B(s2) over the unitsquare [0, 1]× [0, 1]. The last equality is where the time-ordering is accomplished. Inorder to carry out the time-ordering, observe that the unit square [0, 1]× [0, 1] canbe written as

12 | introduction

[0, 1]× [0, 1] ={

(s1, s2) ∈ [0, 1]2 : s1 < s2} ∪ {(s1, s2) ∈ [0, 1]2 : s2 < s1

},

the union of two triangles, up to a set of (�× �)-measure 0 (specifically, the diagonal{(x, x) : x ∈ [0, 1]

}). (Note that the two sets in this union are disjoint.) The operator

product in each term is written so that the operator with the earlier time index actsfirst (to the right). We do this without paying attention to the fact that the operators Aand B do not necessarily commute. Evaluating the integrals leads to the last line.

Example 1.0.2 We stay with the function f (x, y) = xy and operators A,B ∈ L(X).However, we will associate toB a Dirac point mass δτ for τ ∈ (0, 1) and keep Lebesguemeasure � associated with A. Again, to be clear, we can write

A =∫ 1

0A(s) ds and B =

∫ 1

0B(s) δτ (ds),

where A(s) := A and B(s) := B for all 0 ≤ s ≤ 1. Then, denoting the result byf�,δτ (A,B), we obtain, successively,

f�,δτ (A,B) ={∫ 1

0A(s1) ds1

}{∫ 1

0B(s2) δτ (ds2)

}={∫

[0,τ)A(s1) ds1 +

∫(τ ,1]

A(s1) ds1

}{∫{τ}

B(s2) δτ (ds2)}

={∫

{τ}B(s2) δτ (ds2)

}{∫[0,τ)

A(s1) ds1

}

+{∫

(τ ,1]A(s1) ds1

}{∫{τ}

B(s2) δτ (ds2)}

= τBA + (1 – τ)AB.

The reader should notice that the computation above is in the same spirit as that of thefirst example. Indeed, the first equality follows since the measures involved, δτ and �,are probability measures on [0, 1]. The second equality is obtained by writing the firstintegral as an integral over [0, τ) plus an integral over (τ , 1] (since � is continuous,�({τ}) = 0 and thus there is no contribution to the integral from {τ}). The integralfor B can be written as

∫{τ} B(s2)δτ (ds2) since the Dirac measure is supported at {τ}.

Once again, the time-ordering is carried out after the last equality. In the first term,the integral over [0, τ), as it is over times earlier than τ , appears to the right of theintegral of B. In the second term, the integral over (τ , 1] is over times later than τ andso appears to the left of the integral of B. Evaluating the integrals leads to the last line.

Observe that if we allow τ to be one of the endpoints of the closed interval [0, 1], wehave f�,δ0 (A,B) = AB and f�,δ1 (A,B) = BA, for τ = 0 and τ = 1, respectively.

Example 1.0.3 We continue with f (x, y) = xy and the operators A,B ∈ L(X), as inExamples 1.0.1 and 1.0.2. We associate to A a continuous probability measure μ on

introduction | 13

[0, 1] with support in (0, a) and associate to B a continuous probability measure νon [0, 1] with support in (a, 1), where 0< a< 1. Calling the result fμ,ν(A,B), wethen have

fμ,ν(A,B) ={∫

[0,1]A(s)μ(ds)

}{∫[0,1]

B(s)ν(ds)}

={∫

[a,1]B(s)ν(ds)

}{∫[0,a]

A(s)μ(ds)}

= BA.

Once again, the computation proceeds in the same way as in the first two examples.We first write the product AB as a product of integrals, using the fact that the meas-ures involved are probability measures. The second line follows from the fact thatthe measure μ is supported in the interval [0, a] and the measure ν is supported inthe interval [a, 1]. Hence, after the second equality, the integral over [0, a] appearsto the right of the integral over [a, 1]. Evaluating the integrals leads to the last line.Note that, for simplicity, we have assumed that these measures are continuous, i.e.,μ({s}) = ν({s}) = 0, for all s ∈ [0, 1]. (This will be the case throughout this book,with the notable exception of Chapter 8.)

Observe that an entirely analogous computation leads to the (heuristic) result

fν,μ(A,B) = AB.

While the three examples above illustrate the essential ideas of Feynman’s operationalcalculus, it may not be clear how to proceed given an arbitrary function f (z1, . . . , zn) of ncomplex variables. To get some idea of the general disentangling process, we will sketch outhow it works given, say, n noncommuting linear operators. LetA1, . . . ,An be bounded linearoperators on a Banach space X, and associate to each operator Ai a time-ordering measureμi on [0, 1]. This, of course, means that we write

Ai =∫

[0,1]Ai(s)μi(ds),

where Ai(s) := A for 0 ≤ s ≤ 1, i = 1, . . . , n. Assuming that f is analytic on some polydiskP centered at the origin in Cn, we may write

f (z1, . . . , zn) =∞∑

m1, . . . ,mn=0

am1, . . . ,mnzm11 · · · zmn

n .

Let

Pm1, . . . ,mn(z1, . . ., zn) := zm11 · · · zmn

n .

14 | introduction

In view of the examples above, we can compute, using Feynman’s “rules,” the disentan-gling of Pm1, . . . ,mn(A1, . . . ,An) following the time-ordering directions given by the time-ordering measuresμ1, . . . , μn on [0, 1]. One would expect, and this is indeed the case (seeChapter 2), that the disentangled operator f (A1, . . . ,An) would then take the form

f (A1, . . . ,An) =∞∑

m1, . . . ,mn=0

am1, . . . ,mnPm1, . . . ,mn(A1, . . . ,An),

or, to stress the dependence on the time-ordering measures,

fμ1, . . . ,μn(A1, . . . ,An) =∞∑

m1, . . . ,mn=0

am1 , . . . ,mnPm1, . . . ,mnμ1 , . . . ,μn(A1, . . . ,An).

Of course, convergence questions arise, but the series above will converge in operator normon L(X). This series is a perturbation series, in fact a time-ordered perturbation series—each term of the series contains a time-ordered product of powers of the operators Ai.In this, we see definite similarities to the approach to Feynman’s operational calculus viaWiener and Feynman integrals (see [114]).

A natural question at this point is how to make the examples and discussion aboveor, more generally, Feynman’s operational calculus as a whole, mathematically rigorous.One method is to use Feynman and Wiener integrals, along with associated commuta-tive Banach algebras of Wiener functionals called “disentangling algebras,” as was donein Chapters 15–18 of [114]. However, the approach taken in the present book is to de-velop the operational calculus in a rigorous manner starting from Feynman’s heuristic rules.This approach originated with the papers [96–99] by Brian Jefferies and the first author.Indeed, much of [96] is included in Chapter 2, where the basic definitions and propertiesof the operational calculus can be found. We will briefly outline the content here. To followFeynman’s rules in a rigorous way, two commutative Banach algebras, A and D, are con-structed in Section 2.1. To construct the algebra A, we are given n positive real numbersr1, . . . , rn, and we define A := A(r1, . . . , rn) to be the algebra (via pointwise operations) offunctions f (z1, . . . , zn) of n complex variables whose Taylor series

f (z1, . . . , zn) =∞∑

m1, . . . ,mn=0

am1, . . . ,mnzm11 · · · zmn

n ,

centered at the origin in Cn, converges absolutely (and, hence, uniformly as well as point-wise) on the closed polydisk with radii r1, . . . , rn. Such functions are, of course, analytic(i.e., holomorphic) on the corresponding open polydisk with these radii and are continuouson its boundary. A norm ‖·‖A is defined on this algebra, and it is shown that A endowedwith this norm is complete. To obtain the commutative Banach algebra D, we choose nbounded linear operators A1, . . . ,An on a Banach space X and take r1 := ‖A1‖ , . . . , rn :=‖An‖. Furthermore, we attach time indices to each operator Ai, i = 1, . . . , n, via a Borelprobability measure μi on [0,T], T > 0. We let D := D(A1, . . . , An) be the algebra (via

introduction | 15

pointwise operations) obtained by replacing the variables z1, . . . , zn in f (z1, . . . , zn) by the“formal commuting objects” A1, . . . , An. Hence, each element f

(A1, . . . , An

)has the form

f(A1, . . . , An

)=

∞∑m1, . . . ,mn=0

am1, . . . ,mn

(A1)m1 · · · (An

)mn ,

where∥∥ f (A1, . . . , An

)∥∥D

:=∥∥ f∥∥

A<∞. (It is shown in Proposition 2.1.3 that A and D are

isometrically isomorphic.) The commutative Banach algebra D supplies the “commutativeworld” in which Feynman’s originally heuristic rules can be applied in a mathematicallyrigorous way. What we do is to take the Taylor series (centered at the origin)

∞∑m1, . . . ,mn=0

am1, . . . ,mn

(A1)m1 · · · (An

)mn

:=∞∑

m1, . . . ,mn=0

am1, . . . ,mnPm1, . . . ,mn

(A1, . . . , An

)of an element f

(A1, . . . , An

) ∈ D and write it, following Feynman’s rules, as an infinite sumof time-ordered products of powers of the formal objects A1, . . . , An. This is done by firstdisentangling, in D, Pm1, . . . ,mn

(A1, . . . , An

)for every n-tuple (m1, . . . ,mn) of nonnegative

integers; then, once these computations have been completed, the series

∞∑m0, . . . ,mn=0

am1, . . . ,mnPm1, . . . ,mn

(A1, . . . , An

)is a sum of time-ordered expressions in the disentangling algebra D. (The role played bythe time-ordering measures is to specify when operators act in products, and so the time-ordering is done with regard to “directions” supplied by the measures.) Moreover, becauseD is commutative, the time-ordering computations carried out using Feynman’s rules aremathematically rigorous.

Once the time-ordering computations have been done in D, the end result is mappedinto the noncommutative setting of L(X), using the disentangling map Tμ1, . . . ,μn defined inSection 2.2; note the explicit use of time-ordering measures in the notation. We will write

Tμ1, . . . ,μn f(A1, . . . , An

)=

∞∑m1, . . . ,mn=0

am1, . . . ,mnTμ1, . . . ,μnPm1, . . . ,mn

(A1, . . . , An

)for the disentangled operator in L(X). Of course, applying the disentangling computationsterm by term in the Taylor series brings up the question of convergence of the series above.It turns out that this series converges absolutely in operator norm for each f ∈ D (seeProposition 2.2.4). Finally, certain basic properties of the disentangling map are discussedin that section.

16 | introduction

Section 2.3 contains a discussion of several examples of the disentangling process. Alsodiscussed, via examples, is what happens when the time-ordering measures are permuted.Some general observations are made in the course of investigating these examples.

Finally, Section 2.4 discusses what happens with the operational calculus when all of theoperators commute with each other or when a subset of the operators commutes with theremaining operators. The reader will find that these results match nicely with what theirintuition would suggest ought to happen. In particular, ifA1, . . . ,An are a commuting familyof operators, we would expect that

Tμ1, . . . ,μn f(A1, . . . , An

)= f (A1, . . . ,An);

that this is so follows from Proposition 2.4.1.Chapter 3 contains a discussion of several more advanced topics concerning the applica-

tion of the disentangling map defined in Chapter 2. The first topic addressed is the casewhere the Banach space X can be written as a direct sum X = Y ⊕ Z, for some Banachspaces Y and Z, and each operator Ai, i = 1, . . . , n, can be decomposed as Ai = Bi ⊕ Ci forBi ∈ L(Y) andCi ∈ L(Z). As one would expect, it turns out that the disentangled operatorTμ1, . . . ,μnPm1, . . . ,mn

(A1, . . . , An

)can be decomposed as

Tμ1, . . . ,μnPm1, . . . ,mn

(B1, . . . , Bn

)⊕ Tμ1, . . . ,μnPm1, . . . ,mn

(C1, . . . , Cn

);

see Proposition 3.1.1. The second type of disentangling discussed in the first section ofChapter 3 is that of a function f (z1, . . . , zn) which is symmetric in the variables zi1 , . . . , zi� .One might expect the disentangled expression Tμ1, . . . ,μn f

(A1, . . . , An

)to have the same

symmetry. This is indeed the case, as long as a permutation of the operators Ai1 , . . . ,Ai�is accompanied by the same permutation of these operators’ time-ordering measures;see Proposition 3.1.2. The last issue addressed in Section 3.1 concerns analytic functionsg(z1, . . . , zn) which can be written as a tensor product g1 ⊗ · · · ⊗ gn of analytic functionsof one variable. Namely, we explore how these functions behave in the environment of thealgebras A and D; see Proposition 3.1.4.

Section 3.2 discusses how disentanglings can be computed using time-ordering measuresthat are not probability measures. Though it is often most convenient to use probabilitymeasures as time-ordering measures, in certain cases it is desirable to use time-orderingmeasures that are not probability measures. For instance, in the study of evolution prob-lems for the operational calculus (see Chapter 6), time-ordering measures that are notprobability measures will be needed. Hence, it is necessary to determine how the oper-ational calculus is affected when nonprobability measures are used in the computationof a disentangling. Since the disentangling map is defined in Chapter 2 using probabil-ity measures for the time-ordering measure, what is done to define the disentanglingmap using nonprobability measures is to first “normalize” the measures using the totalvariation norm. Then, we use these normalizations to carry out the disentangling, andfinally multiply the disentangling by appropriate powers of the total variation norms of

introduction | 17

the measures. In fact, if A1, . . . ,An ∈ L(X) and we associate to each Ai a nonzero Borelmeasureμi on [0, 1], we obtain

Tμ1, . . . ,μnPm1, . . . ,mn

(A1, . . . , An

)= ‖μ1‖m1 · · · ‖μn‖mn Tμ1/‖μ1‖, . . . ,μn/‖μn ‖P

m1, . . . ,mn(A1, . . . , An

).

From this, it follows that the form of the disentangling does not change when the time-ordering measures are not probability measures.

Section 3.3 addresses a natural way of simplifying certain disentangling computations.Given operators A1, . . . ,An and associated time-ordering measures μ1, . . . , μn, it is as-sumed that the supports S(μj) of the time-ordering measures have a particular order. If,for example, the supports S(μj) are such that

S(μ1) ≤ S(μ2) ≤ · · · ≤ S(μn),

where S(μ) ≤ S(ν) means that the support ofμ lies to the left of the support of ν, we wouldthen expect that the disentangling of the monomial Pm1, . . . ,mn

(A1, . . . , An

) ∈ D under thedirections given by the measuresμ1, . . . , μn would take the form

Amnn · · ·Am2

2 Am11 .

A moment’s reflection will convince the reader that this is reasonable. Since the support ofthe time-ordering measure μ1 associated to the operator A1 lies to the left of (or is earlierthan) the supports of all of the other measures, it stands to reason that all factors of A1 willact first in products (i.e., on the right). Then, because the support of the measure μ2 asso-ciated to A2 is to the right of (or later than) that ofμ1 and to the left of (or earlier than) theremaining measures, we expect that the factor acting immediately after the factor Am1

1 willbeAm2

2 . This process continues, and we do indeed obtain the disentangling indicated above.Of course, the case just outlined is the easiest case to deal with, but the ideas containedin the more complex cases addressed in Section 3.3 are similar to those just described.It is worth noting that the idea of ordering of the supports of the time-ordering meas-ures is a recurring one which will be encountered several times in this book, especially inChapters 4 and 5.

The last section of Chapter 3, Section 3.4, discusses the idea of disentangling so-calledexponential factors. (We note that Feynman used the term “experimental factors”; see[58].) In particular, the time independent version of these exponential factors is investi-gated here. (This will be revisited in much greater generality in Chapter 6, especially inSections 6.2 and 6.3.) What, then, do we mean by “exponential factor”? Let B1, . . . ,Bk ∈L(X) and let A ∈ L(X). Given nonnegative integers m, n1, . . . , nk, an exponential factor(in the disentangling algebra D) is

AmBn11 · · · Bnk

k .

18 | introduction

What is done is to compute the disentangling of AmBn11 · · · Bnk

k , while keeping track of whereA appears in products. As one might expect, the end result of the disentangling will, veryinformally, look like∑

(coefficient)Apower(product of factors of Bi′s)Apower · · ·

(product of factors of Bi′s)Apower.

(See the discussion preceding Theorem 3.4.1; see also the results that follow this theorem.)One reason why we track the occurrences of the operatorA is that this operator will later bereplaced by the generator of a (C0) semigroup of linear operators (see Chapter 6).

Chapter 4 begins by addressing the application of the operational calculus to the fol-lowing situation. Suppose that we have operator–measure pairs (A1,μ1), . . . , (An,μk) forwhich the supports of the measuresμ1, . . . , μk are contained in the interval [0, a] ⊂ [0,T](0 < a < T). Suppose also that the operator–measure pairs (Ak+1,μk+1), . . . , (An,μn) aresuch that the supports of the measures are contained in [a,T]. Now, consider how todisentangle the expression

Pm1, . . . ,mk ,mk+1, . . . ,mn(A1, . . . , Ak, Ak+1, . . . , An)

:= Am11 · · · Amk

k Amk+1k+1 · · · Amn

n

= Pm1, . . . ,mk(A1, . . . , Ak)Pmk+1, . . . ,mn(Ak+1, . . . , An).

Since the supports S(μj), j = 1, . . . , k, are all to the left of (or earlier than) the supportsS(μj), j = k + 1, . . . , n, we would expect that the disentangling would be

Tμ1, . . . ,μk ,μk+1, . . . ,μnPm1, . . . ,mk ,mk+1, . . . ,mn(A1, . . . , Ak, Ak+1, . . . , An)

= T μk+1, . . . ,μnPmk+1, . . . ,mn(Ak+1, . . . , An)Tμ1, . . . ,μkP

m1, . . . ,mk(A1, . . . , Ak).

With a little thought, we should to be able to write this expression as

Tμ1, . . . ,μk ,μk+1, . . . ,μnPm1, . . . ,mk ,mk+1, . . . ,mn(A1, . . . , Ak, Ak+1, . . . , An)

= Tμ0,μk+1, . . . ,μnP1,mk+1, . . . ,mn

(Km1, . . . ,mk , Ak+1, . . . , An

)=[Tμk+1, . . . ,μnP

mk+1, . . . ,mn(Ak+1, . . . , An

)]Km1, . . . ,mk ,

where μ0 is any (continuous) probability measure supported in [0, a] and Km1, . . . ,mk :=Pm1, . . . ,mk

(A1, . . . , Ak

), andKm1, . . . ,mk is the disentangled operator we obtain from Km1, . . . ,mk .

The operator Km1, . . . ,mk is called a “linear factor” and the process that leads to the disentan-gling above is called “extracting a linear factor.” The function being disentangled after thesecond equality above is

ζ zmk+1k+1 · · · zmn

n ;

introduction | 19

ζ is a linear factor. While the idea of extracting a linear factor is intuitively rather clear, acareful proof of the corresponding result (Theorem 4.1.1) is fairly involved, as it dependson many of the basic results and definitions relating to the disentangling process. Followingthis theorem, a number of related results and examples are considered.

The second section of Chapter 4 is, to a degree, similar to the first section. However,instead of having one cluster of time-ordering measures supported in the interval [0, a],we will have two clusters of measures: one cluster, μ1, . . . , μk, supported in [a, b],and another cluster of measures, μk+1, . . . , μk+l, supported in [c, d], where a < b <c < d < T. Finally, the remaining measures, μk+l+1, . . . , μn, are supported in [0, a] ∪[b, c] ∪ [d,T]. Ideas similar to those used to extract a linear factor in Section 4.1 en-able us to disentanglePm1, . . . ,mn

(A1, . . . , An

)by first disentanglingPm1, . . . ,mk

(A1, . . . , Ak

)and

Pmk+1, . . . ,mk+l(Ak+1, . . . , Ak+l

), thereby obtaining the operators Km1, . . . ,mk and Lmk+1, . . . ,mk+l ,

respectively. We then compute

Tμ0,ν0,μk+l+1, . . . ,μnP1,1,mk+l+1, . . . ,mn

(Km1, . . . ,mk , Lmk+1, . . . ,mk+l , Ak+l+1, . . . , An

),

where μ0 is any probability measure supported in [0, a] and ν0 is any probability measuresupported in [c, d]. The function being disentangled here is

ζ1ζ2zmk+l+1k+l+1 · · · zmn

n ;

the term ζ1ζ2 is called a bilinear factor.In the third section of Chapter 4, Section 4.3, the methods and results from the first two

sections are extended to carry out the task of extracting a multilinear factor. The notationis more involved in this section, but the essential ideas are similar to those used to extractlinear and bilinear factors.

Section 4.4 also uses the essential ideas of the first two sections; however, multilinearitywill not be present in the process. We again carry out the disentangling process accord-ing to how the clusters of time-ordering measures are supported. However, it will notbe the case in this section that the disentangled operators will be “factored out” as wasdone when linear, bilinear and multilinear factors were considered. More specifically, leta1, . . . , ad, b1, . . . , bd be real numbers satisfying

0 ≤ ad ≤ ad–1 ≤ · · · ≤ a2 ≤ a1 < b1 ≤ b2 ≤ · · · ≤ bd–1 ≤ bd ≤ T.

We assume that, for some subset I1 of {1, . . . , n}, the cluster {μi : i ∈ I1} of time-orderingmeasures is supported in [a1, b1]. Next, we assume that for some subset I2 of {1, . . . , n},the cluster {μi : i ∈ I2} of time-ordering measures is supported in [a2, a1] ∪ [b1, b2]. Wecontinue the process to obtain subsets Ij, j = 1, . . . , d, of subsets of {1, . . . , n} for which thecluster

{μi : i ∈ Ij

}of time-ordering measures is supported in [aj, aj–1] ∪ [bj–1, bj] (where

a0 := b1 and b0 := a1). Now, let L0 be the identity operator, and define the monomial

P1;mi ,i∈I1(L0; Ai, i ∈ I1)

20 | introduction

to be the monomial in D which consists of the product of L0 with factors Amii , i ∈ I1. We

then disentangle this monomial to obtain the operator

L1 := Tη0;μi ,i∈I1P1;mi ,i∈I1(L0; Ai, i ∈ I1

);

this is the disentangling corresponding to the cluster of time-ordering measures supportedin [a1, b1], where η0 is any continuous probability measure supported in [a1, b1]. With thedisentangled operator L1 in hand, we define L2 to be the operator obtained by disentan-gling the monomial with L1 appearing to the first power (and associated to a continuousprobability measure η1 supported in [a1, b1]), and with the operators Ai, i ∈ I2, with expo-nents mi, i ∈ I2, and associated time-ordering measures μi, i ∈ I2, which are supported in[a2, a1] ∪ [b1, b2]. The reader will note that the measures μi, i ∈ I2, are supported bothto the left and to the right of the measures μi, i ∈ I1 (which are supported in [a1, b1]).Therefore, the operator L1 cannot be extracted from the disentangling that determines theoperator L2, which is defined as

L2 := Tη1;μi ,i∈I1P1;mi ,i∈I2(L1;Ai, i ∈ I2).

This disentangling process continues, working outward through the unions [aj, aj–1]∪ [bj–1, bj], and we obtain the iterative collection L1, L2, . . . , Ld of operators given by

Lj := Tηj–1;μi ,i∈IjP1;mi ,i∈Ij(Lj–1;Ai, i ∈ Ij

),

for j = 1, . . . , d, where the continuous probability measures ηj are supported in [aj, bj].Section 4.6 discusses the idea of decomposing disentangling. Suppose we have continu-

ous probability measuresμ1, . . . , μn on [0, 1], and let t ∈ (0, 1). Writeμj,1 := [0, t].μj andμj,2 := [t, 1].μj, j = 1, . . . , n. (If λ is a measure and if U is a λ-measurable set, we denotethe restricted measure V �→ λ(U ∩ V) by U.λ in Section 4.6.) For example, we would ex-pect to be able to write, given these decompositions of the time-ordering measures, thedisentangling of exp

(A1 + · · · + An

)over the interval [0,T] as

Tμ1, . . . ,μn eA1+ · · · +An = Tμ1,2, . . . ,μn,2e

A1+ · · · +AnTμ1,1, . . . ,μn,1eA1+ · · · +An .

This is a decomposition of the disentangling of the exponential function over the inter-val [0,T] into a product of the disentanglings of the same function over the intervals[0, t] and [t,T]. That this is in fact true follows from Theorem 4.6.7.

Moving on to Chapter 5, we note that the first section introduces two noncommutat-ive operations (a noncommutative multiplication, ⊗, and a noncommutative addition,

◦+)

that serve to relate the “full” disentangling algebra D(A1, . . . , An

)to (at least) two other

disentangling algebras in the following way. (We will discuss the simplest case here,for illustrative purposes.) Suppose that the measures μ1, . . . , μk associated to the oper-ators A1, . . . ,Ak are supported in [0, s] for 0< s< t, and that the measures μk+1, . . . , μn

introduction | 21

associated to the operators Ak+1, . . . ,An are supported in [s, t]. We can then constructthe two disentangling algebras D

(A1, . . . , Ak

)and D

(Ak+1, . . . , An

), determined by the

supports of the time-ordering measures associated with the respective sets of operators.The first algebra will be denoted by Dt and the second algebra by Ds,t . Two noncommut-ative operations ⊗ and

◦+ are defined, which take f ∈ Ds, g ∈ Ds,t and give us functions

f ⊗ g, f◦+ g ∈ D

(A1, . . . , An

). In other words, the operations ⊗ and

◦+ map pairs of dis-

entangling algebras into a larger disentangling algebra. We will see that these operationsare linear and continuous in both arguments, and it will also be verified that they satisfyall of the expected algebraic properties. The second section of the chapter provides a se-lection of examples using these noncommutative operations. In the third section of thechapter, the relation between the operations ⊗ and

◦+, the disentangling map Tμ1, . . . ,μn ,

and the disentangling maps Tμ1, . . . ,μk and Tμk+1, . . . ,μn is demonstrated—see Theorem 5.3.1and Corollary 5.3.4. Furthermore, a binomial and multinomial formula for the operation◦+ and an exponential formula exp

(f1

◦+ · · · ◦+ fn

)= exp ( f1)⊗ · · · ⊗ exp ( fn) are estab-

lished. It is also noted that these operations enable the computation of expressions suchas cos

(f◦+ g)

and sin(f◦+ g)

.It is in Chapter 6 that we enter into a discussion of the operational calculus in the

presence of time-dependent operators (i.e., operator-valued functions). As the reader willnotice, there is a marked correspondence between the key definitions and properties ofthe operational calculus in the time-dependent setting and those for the time independentsetting found in Chapters 2–5.

In Section 6.1, the time-dependent operational calculus is developed. Instead of fixedoperators A1, . . . ,An from L(X), we instead have L(X)-valued functions Ap : [0,T] →L(X), p = 1, . . . , n. (Here, T > 0 is fixed.) Associated with each operator-valued functionAp(·) is a continuous Borel probability measure μp on [0,T] (a time-ordering measure).Owing to the time dependence, we will find it necessary to impose certain measur-ability/integrability conditions on these functions in order for integrals of time-ordered(operator) products of these functions to make sense. It also happens that the presenceof time dependence causes the definition of the commutative Banach algebras A and D todepend explicitly on the time-ordering measures. This is different from the time independ-ent setting of Chapter 2, where the definition of the algebras A and D depends only onthe operator norms of each operator. Nevertheless, A and D continue to be isometricallyisomorphic in the time-dependent setting. Furthermore, it turns out that the definition ofthe disentangling map in the time-dependent setting is exactly the same as the one given inChapter 2. This may be surprising, but once the reader compares the relevant definitionsin Chapter 2 with those in Section 6.1, it will be clear why this is the case. However, onedifference is notable—the fact that, while the disentangling map continues to be a contrac-tion in the time-dependent setting, it may no longer be a norm-one contraction, as in thetime independent setting. Furthermore, the presence of time-dependent operators does, attimes, cause changes in the conclusions of theorems and tends to complicate the proofs ofsome results.

22 | introduction

Section 6.2 presents an extensive discussion of the disentangling of the exponentialfunction

exp{

–tα +∫ t

0β1(s)μ1(ds) + · · · +

∫ t

0βn(s)μn(ds)

},

where α is an unbounded operator (the generator of a strongly continuous semigroup ofbounded linear operators onX: in short, a (C0) semigroup of linear operators) and, for eachp = 1, . . . , n, the function βp(·) is an operator-valued function, i.e., a time-dependent oper-ator. (The content of Section 6.2 first appeared in the paper [34] written by Brian DeFacioand the first two authors of this volume, and can also be found, in essence, in [114, Chapter19].) While this section is rather lengthy, its primary result, the evolution equation that isthe subject of Theorem 6.2.11, will play a crucial role later in Chapter 6. The disentanglingof the exponential function above, in the presence of the unbounded operator α, is carriedout by applying, in a heuristic way, Feynman’s rules that track where the operatorα appearsin time-ordered products of operator-valued functions. The reason for taking a heuristicapproach to the disentangling is that the rigorous approach introduced in Chapter 2 can-not, at the current stage of development of the theory, accommodate unbounded operators.However, even though the disentangling computations are heuristic in nature, the result-ing infinite series of integrals of time-ordered products of operator-valued functions turnsout to converge in operator norm. Furthermore, the disentangled exponential function isshown to be the unique solution to the evolution equation obtained in Theorem 6.2.11.This integral equation is easily seen to be equivalent to certain standard partial differen-tial equations (the heat equation, the Schrödinger equation, etc.) for specific choices ofsemigroup generators and operator-valued functions.

Found in Section 6.3 is a discussion of the disentangling of exponential factors in thetime-dependent setting. (Section 3.4 discusses the disentangling of exponential factors inthe time independent case.) The reason we disentangle exponential factors in the time-dependent setting arises from our wish to develop a rigorous version of the disentangledexponential function. We are able to use the disentangling of exponential factors in thetime-dependent setting to define a version of the disentangling map in the presence ofthe (typically) unbounded generator of a (C0) semigroup. As mentioned above, the dis-entangling of this exponential function involves keeping track of where the unboundedoperator α appears in operator products. The idea of tracking when a particular operatorappears in products is applied first to monomials ArBm1

1 · · ·Bmnn ; we wish to keep track of

the operator A. (The operators Bp are taken here to be time-dependent, and A is takento be time independent.) The disentangling of the aforementioned monomial, keepingtrack of the operator A, leads us in turn to a natural definition of the disentangling of thefunction ez0 f (z1, . . . , zn), where z0 is replaced with the generator of a semigroup of linearoperators. This definition allows us to obtain, in a very natural way, the disentangling of theexponential function

exp{

–tα +∫ t

0β1(s)μ1(ds) + · · · +

∫ t

0βn(s)μn(ds)

}

introduction | 23

that was derived in Section 6.2 by using heuristic considerations as the starting point for thecomputation. In this section, however, the disentangling is obtained using the disentanglingmap approach. It is also possible to use the approach presented in Section 6.3 to disentan-gle a rather broad class of functions that involve unbounded generators of semigroups ofoperators.

In Section 6.4, the evolution equation of Theorem 6.2.11 is used to obtain a general-ized integral equation for the operational calculus in the presence of the generator of a (C0)semigroup. This generalized integral equation will allow us to describe the time evolutionof a disentangled operator that is different from the disentangled exponential function ofSection 6.2. What allows us to obtain the integral equation is an observation about therelation between the disentangled exponential function of Section 6.2 and functions ofthe form

f (z0, z1, . . . , zn) = ez0g(z1, . . . , zn).

As we note in Section 6.4, the difference between the disentangling of a functionh(z1, . . . , zn) and the exponential function exp(z1 + · · · + zn) lies in the difference betweenthe Taylor coefficients in the series expansion of h(z1, . . . , zn) at 0 ∈ Cn and the seriesexpansion of exp(z1 + · · · + zn) at 0 ∈ Cn. We effect the change of coefficients via theuse of Cauchy’s integral formula for derivatives. It turns out, however, that the disentan-gling which is obtained via Cauchy’s formula has to be modified somewhat to achievecompatibility with the evolution equation found in Theorem 6.2.11. The modified dis-entangling is referred to as a “reduced” disentangling (“reduced” in the sense that theseries that describes the disentangling contains fewer terms than the disentangling thatresults from the standard definitions of Chapter 2 and Section 6.1). It is the reduced dis-entangling that gives rise to the generalized integral equation in Theorem 6.4.2. Using thisintegral equation, it is shown, via an example, that the reduced disentangling supplies so-lutions to the heat equation. Furthermore, again using Theorem 6.4.2, we outline somerelations between the reduced disentangling and the analytic-in-time and analytic-in-massoperator-valued Feynman integrals discussed in Sections 13.2 and 13.5–13.6 (along withSection 15.1) of [114].

In Chapter 7, we turn to an investigation of the stability, or continuity, properties of theoperational calculus. There are three types of stability that are considered in this chapter:stability of the operational calculus with respect to the time-ordering measures, stability ofthe operational calculus with respect to the operators (or operator-valued functions in thetime-dependent case), and joint stability, that is, stability with respect to both the operators(or operator-valued functions) and the time-ordering measures. Before continuing, we willtake the time to comment briefly on what is meant by stability in each case, leaving the moredetailed discussion for Chapter 7.

Suppose that, given operators A1, . . . ,An ∈ L(X), we have associated to each oper-ator Ai a time-ordering measure μi (a continuous Borel probability measure on [0,T]).We select, for each i = 1, . . . , n, a sequence {μi,k}∞k=1 of continuous Borel probabilitymeasures on [0,T] which converges weakly to the measure μi. (Here, “weak conver-gence” is to be understood in the probabilistic sense. To a functional analyst, weak

24 | introduction

convergence is weak-∗ convergence.) By stability in this context, we mean that the sequence{Tμ1, k , . . . ,μn, k

}∞k=1

of disentangling maps converges in some suitable sense to an element ofL(D,L(X)). In fact, it turns out that the sequence

{Tμ1, k , . . . ,μn, k

}∞k=1

converges in the strongoperator topology on L(D,L(X)); i.e., for each f ∈ D,

limk→∞

Tμ1, k , . . . ,μn, k f(A1, . . . , An

)= Tμ1, . . . ,μn f

(A1, . . . , An

).

Another way to look at this type of stability is to observe that each n-tuple (μ1, k, . . . , μn, k),k ∈ N, determines a particular operational calculus. The sequence

{Tμ1, k , . . . ,μn, k

}∞k=1

istherefore a sequence of operational calculi, and the existence of a limit means that thissequence of operational calculi has a limiting operational calculus. Even though this dis-cussion concerns the time independent setting, the basic ideas outlined above can also beapplied in the time-dependent setting.

The second type of stability is with respect to the operators (or operator-valued func-tions, in the time-dependent setting). Once again, we will discuss the time independentsetting here. The ideas are very similar in the time-dependent setting. Thus, we fix an n-tuple(μ1, . . . ,μn) of time-ordering measures and choose sequences {A1, k}∞k=1 , . . . , {An, k}∞k=1 ofoperators that converge, in some sense, to the operators A1, . . . ,An, respectively. Giventhese sequences of operators, we have, for any f ∈ D, a corresponding sequence of dis-entangled operators,

{Tμ1, . . . ,μn f (A1, k, . . . ,An, k)

}∞k=1, in L(X). We ask if the convergence

of the sequences of operators implies the convergence of the disentangled operators. Notethat the choice of an n-tuple of time-ordering measures fixes the operational calculus, andthe convergence of the disentangled operators then takes place within this operationalcalculus.

The final type of stability considered in Chapter 7 is joint stability. Here, for eachj = 1, . . . , n, we not only choose sequences

{μj, k

}∞k=1 of measures converging weakly to μj

for each j, but also choose sequences{Aj, k}∞k=1 of operators converging appropriately to an

operator Aj. For each f , we then have a sequence{Tμ1, k , . . . ,μn, k f

(A1, k, . . . , An, k

)}∞k=1

,

which, we hope, will converge in some fashion to an operator Tμ1, . . . ,μn f(A1, . . . , An

).

In this setting, as above, each n-tuple (μ1, k, . . . , μn, k) determines a particular oper-ational calculus and, in each of these calculi, we have the corresponding operatorTμ1, k , . . . ,μn, k f

(A1, k, . . . , An, k

), with the arguments of f also depending on the sequential in-

dex k. So, for each value of k ∈ N, we are selecting an element of the operational calculusindexed by the n-tuple (μ1, k, . . . , μn, k).

A brief outline of the four sections of Chapter 7 follows. Section 7.1 is concerned withthe general setting for the stability theory. It is here that the reader will find the neces-sary definitions and theorems which form the underlying structure for the later sections.In Section 7.2, we investigate joint stability in both the time-dependent and the timeindependent setting. The theory of stability with respect to the operators (or operator-valued functions) is presented in Section 7.3, again in both the time-dependent and the

introduction | 25

time independent setting. Finally, Section 7.4 presents the theory of stability with respect tothe time-ordering measures in both the time-dependent and the time independent setting.

Moving on to Chapter 8, we change a fundamental aspect of the operational calcu-lus. Specifically, instead of using continuous measures as time-ordering measures that areassociated to the operators involved, Chapter 8 develops the operational calculus usingtime-ordering measures that have a nontrivial discrete part. In particular, to each operatorAj, j = 1, . . . , n, we associate a measure λj = μj + ηj, where μj is a continuous measure andηj is a purely discrete, finitely supported, measure. We find that allowing time-orderingmeasures with nonzero discrete parts gives rise to significant combinatorial difficulties inthe general setting and also in some special cases that are of interest. Also worth notingis that, while we could consider allowing the discrete parts of the time-ordering measureshave countably infinite support, the combinatorial difficulties would likely be overwhelm-ing. Fortunately, we find that most of the questions we consider, including some which arephysically motivated, can be dealt with using simpler assumptions about the ηj’s which wewill make in this chapter.

Recall that a Borel measure μ on R is continuous (or purely continuous) if μ ({x}) = 0for all x in the support S(μ) of μ. Furthermore, a Borel measure η on R is discrete (orpurely discrete) if η =

∑∞p=1 wpδτp , where wp ∈ C and τp ∈ C, and where δτp denotes the

Dirac measure with total mass one concentrated at{τp}

. The support S(η) of η is thenequal to

{τp : p ≥ 1

}. A finitely supported discrete measure η is a discrete measure for which

S(η) is finite (or, equivalently, for which wp = 0 for all sufficiently large p). Finally, it is wellknown that every Borel measure λ on R can be uniquely written as the sum of a continuousmeasure μ and a (not necessarily finitely supported) discrete measure η: λ = μ + η; see,for example, [26] or [187].

The structure of Chapter 8 is much like that of Chapter 2 (and of Section 6.1). We firstdeal with the time-ordering of the monomial

Pm1, . . . ,mn(A1, . . . , An

)=(A1)m1 · · · (An

)mn .

As was true in Chapter 2, the time-ordering of this monomial is crucial to making the con-nections between Feynman’s ideas and the rigorous approach developed in this book. Wethen move on to define the disentangling of an arbitrary element f of the disentangling alge-bra, giving rise to the definition of the disentangling map in this more complicated setting.It turns out, just as in Chapter 2, that the disentangling map is a norm-one contraction inthe time independent setting, but is not necessarily of norm one in the time-dependent set-ting (see Section 6.1). As one would expect, the proof of the continuity of the disentanglingmap in the setting of this chapter is much more difficult. Sections 8.1–8.3 serve to developthese ideas in the combined continuous/discrete setting.

In Section 8.4, we apply the disentangling map in the combined setting to compute anumber of special cases that are of interest.

Section 8.5 turns to the consideration of some stability results. These results are in thespirit of those which we proved in Chapter 7, but, as is the case for most of this chapter,the proofs are complicated significantly by the presence of measures with nonzero dis-crete parts. The reader will find that the stability results of Section 8.5 are not as general as

26 | introduction

those obtained in Chapter 7. This is due to the complications introduced by time-orderingmeasures that have nonzero discrete parts. However, we are able to establish stabilitywith respect to the time-ordering measures in both the time independent and the time-dependent settings when each time-ordering measure is a finitely supported purely discretemeasure and when we have a combination of continuous and purely discrete time-orderingmeasures. Stability with respect to the operators (or operator-valued functions) is estab-lished quite generally, with time-ordering measures that have both continuous and finitelysupported discrete parts.

Chapter 9 explores a differential or derivational calculus which can be associated withthe disentangled operators arising from the approach to the operational calculus presentedin this book. The most important part of Chapter 9 deals with a first-order calculus for ananalytic function of n noncommuting variables.

In Section 9.2, the relations between the disentangling map, homomorphisms and an-tihomomorphisms are investigated. (Recall that, given Banach algebras A and B, thelinear map φ : A→ B is an antihomomorphism if φ(xy) = φ(y)φ(x) for all x, y ∈ A.)In particular, if φ : L(X) → L(Y) is a continuous homomorphism of algebras, it is

shown that, for appropriate analytic functions f , we have Tμ1, . . . ,μn f(φ(A1), . . . , φ(An)

)=

φ(Tμ1, . . . ,μn f

(A1, . . . , An

)). The same relation is also established for an antihomomorph-

ism ψ : L(X) → L(Y). Finally, given a continuous linear operator H : X → Y such thatAjH = HBj for j = 1, . . . , n, it is shown that H commutes with the disentangling map.

Section 9.3 establishes first-order expansions by making use of the so-called derivationformula: If f ∈ D and if D is an arbitrary derivation of L(X), then

D[Tμ1, . . . ,μn f

(A1, . . . , An

)]=

n∑j=1

Tμ1, . . . ,μj ,μj ,μj+1, . . . ,μnFj(A1, . . . , Aj, D(Aj), Aj+1, . . . , An

),

where Fj(x1, . . . , xj, y, xj+1, . . . , xn) := (∂/∂xj)f (x1, . . . , xn)y andμ1, . . . , μn are continuousprobability measures. This formula is used to determine an expression for the operator C1

in the expansion

f (A + εB) = f (A) + εC1 + ε2C2 + · · · + O(εn) as ε → 0+.

We will find that C1 = Tμ,μf1(A, B).With the derivation formula of Section 9.3 established, some special cases of higher-order

expansions are provided in Section 9.4. A Maclaurin expansion with remainder and a Taylorexpansion with remainder are obtained.

Turning to Chapter 10, we note that the chapter begins with a sketch of the necessaryelements of the theory of distributions (or generalized functions). This material is stand-ard and can be found in many references. However, we follow the presentation given byHörmander [87]. Of particular importance for us in Chapter 10 is the Paley–Wiener–Schwartz theorem (Theorems 10.2.16 and 10.2.17) on the representation of distribu-tions of compact support using the Fourier transform. The main section of Chapter 10,

introduction | 27

Section 3, gives a proof that for self-adjoint operators A1, . . . ,An on a Hilbert space, then-tuple (A1, . . . ,An) is of Paley–Wiener type (0, r, μ) for some r > 0 and any n-tupleμ = (μ1, . . . , μn) of continuous Borel probability measures on [0, 1] (see Definition10.3.1). Of course, if the n-tuple (A1, . . . ,An) is of Paley–Wiener type, then, using thevector-valued Paley–Wiener–Schwartz theorem, we are able to obtain a rich extension ofthe functional calculus f �→ fμ(A) from analytic functions to functions that are C∞ in aneighborhood of the (compact ) support γμ(A) of the operator-valued distribution

Fμ,A(f ) := (2π)–n∫

RnTμ1, . . . ,μn

(eı〈ζ ,A〉

)f (ζ ) dζ .

The chapter ends with a discussion of various Trotter-like product formulas that can beobtained in the Banach space setting.

Finally, in the epilogue to this book, Chapter 11, we present several open problems andsuggestions for future research connected to various aspects of the theory of Feynman’soperational calculus developed in this monograph. The problems discussed involve pos-sible extensions of the theory to noncommuting unbounded linear operators, general Borelmeasures (possibly with countably supported discrete parts) and a search for the corres-ponding evolution equations. They also deal with a variety of topics in mathematics andphysics, including noncommutative (or free) probability theory and combinatorics, quan-tum statistics (such as Bose–Einstein and Fermi–Dirac statistics), quantum field theory andrenormalization, abstract path integrals, and a possible extension of the operator calculus tothe context of operator algebras (such as Banach algebras, C∗-algebras and von Neumannalgebras). The final open problem (Problem 11.2.11) is connected in many different waysto most of the other open problems presented in the epilogue, as well as to various otheraspects of this book (and of the concrete approach to Feynman’s operational calculus de-veloped in [114, Chapters 14–18]). It involves a possible (and still largely hypothetical)algebraic and combinatorial framework involving disentangling algebras, now viewed asHopf algebras or quantum groups and their categorical analog. This research direction isstill mostly unexplored. (However, as is briefly mentioned at the end of Chapter 11, thesecond author and one of his students, Dominick Scaletta, have begun investigating it in[146], where it was shown that a natural quantum group structure can be associated withthe disentangling algebras and the corresponding disentangling maps discussed in the ab-stract theory developed in this book, and similarly for the associated disentangling algebrasand the corresponding analytic Feynman integrals of [110, 113, 114].) We invite interestedreaders to pursue this direction wherever their imagination may lead them, as well as todevelop their own problematics and their own approach to this rich and fascinating subject.

We close this introduction with some short remarks concerning work that has alreadybeen done with Feynman’s operational calculus and related topics. We organize the ref-erences into four rather broad categories: (i) work that was done, primarily before 1980,concerning operator calculus, with an eye towards the use of Feynman’s ideas concern-ing operator calculus; (ii) work done since 1980 involving the use of product formulasand functional integrals; (iii) work done, again since 1980, that makes use of Feynman’sheuristic rules to obtain results that are then rigorously verified; and (iv) work done since

28 | introduction

1999 within the abstract setting of this monograph (i.e., work that is based on the originalpapers by Brian Jefferies and the first author).

As one would expect, the boundaries between these categories are blurred and there isbound to be some significant overlap between them. Furthermore, while a fair number ofreferences are provided, the lists of references below are certainly not complete in any wayand are meant only to provide a sample of existing work. (Moreover, a number of additionalrelated references are provided in the epilogue to this book, Chapter 11.) However, the mo-tivated reader can delve into the bibliographies of the references listed to find other workson Feynman’s operational calculus and related areas. The references are as follows:

(i) First, we take note of early work done with the operator calculus, nearly all ofwhich was published before 1980. The following references make use of func-tional integrals (i.e., Wiener or Feynman integrals) or integrals of time-orderedproducts of operators to form functions of several noncommuting operators.Using these techniques, some related differential or evolution equations weredeveloped. These publications are [7, 21, 35, 45–48, 59, 68, 154, 155, 158, 163,164, 213]. In particular, the book Operational Methods by Maslov [154] and thepapers by Nelson [164] and Araki [7] were influential. (A later book, Methods ofNoncommutative Analysis by Nazaikinskii, Shatalov and Sternin [162], extendedand developed further many of the ideas of Maslov.)

Other work that is related to the operational calculus (and, in particular, theformation of functions of several operators; i.e., operator calculus) can be foundin the references [4, 6, 22–25, 27, 28, 37–40, 42, 43, 60–65, 84–91, 94, 95, 129,180, 204, 205]. We note that several of these references do not explicitly refer toFeynman’s work on the operator calculus. Also, it is worth mentioning that time-ordered integrals appeared in the mathematical literature quite some time beforeFeynman’s 1951 paper [58]. Indeed, time-ordering in integrals can already befound in the work of Vito Volterra [209]; see also [210].

(ii) The next group of references, published since 1980, addresses, at least in part,product formulas, path integrals and their use in Feynman’s operational calcu-lus and in the study of Feynman integrals. The corresponding references are[1, 2, 12, 19, 106, 107, 110–114, 125, 132–143, 157, 191]. These references in-clude the 2000 bookThe Feynman Integral and Feynman’sOperational Calculus byJohnson and Lapidus [114], the second part of which (Chapters 14–19) is dedi-cated to the rigorous development of Feynman’s operator calculus via Wienerand Feynman path integrals, generalized Dyson (or perturbation) series, andassociated disentangling algebras (Chapters 15–18, based in particular on [110–113, 137–143]) or via more abstract operator methods (Chapter 19, based on[33, 34]). See the Preface to the present book for a brief discussion of the rele-vant part of [114] and its relationship to the approach to Feynman’s operationalcalculus developed in this volume, and see also part (iii) of this list.

We also mention here the books [77, 90, 197, 201, 202, 211, 212], wherequantization via functional integrals is used in quantum mechanics and quantumfield theory. (Many other relevant references could certainly be provided on this

introduction | 29

and related subjects.) The book [199] focuses on the lives of Dyson, Feynman,Schwinger and Tomonaga as well as on the development of quantum electro-dynamics by these physicists. Parts of the interesting biographies of Feynmanby Gleick [76] and Mehra [156], as well as of Schweber’s expository article[198], also discuss aspects of Feynman’s operational calculus from a historicalperspective.

Work related to these references can be found in [85, 86, 94, 103].(iii) The references, also (for the most part) published after 1980, that are oriented

towards making Feynman’s operational calculus mathematically rigorous but donot make use of the abstract approach developed in this monograph deal withthe use of Feynman’s heuristic rules to derive results that are then rigorously veri-fied. These papers and books are [33, 34, 69–75, 110, 114, 139–142, 189]. Otherwork which addresses aspects of operator calculus but is not directly related toFeynman’s operational calculus can be found in [4, 28, 32, 95, 103, 104, 126–128, 181–185, 190].

(iv) We now turn to citing work that has been carried out in the abstract setting of thismonograph, i.e., papers that have been written since the original papers [96–99]by Jefferies and the first author. This work has been carried out since 1999 andcan be found in [3, 100–102, 108, 109, 115, 116, 166–175].

Finally, closely related work concerning the abstract formulation of the oper-ational calculus can be found in [18, 112, 113, 143, 146, 178].