c. wiesendanger- quantum isometrodynamics

8/3/2019 C. Wiesendanger- Quantum Isometrodynamics

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arXiv:0903

.2625v1[math-ph]15Mar2009

QUANTUM ISOMETRODYNAMICS

C. WIESENDANGER

Aurorastr. 24, CH-8032 ZurichE-mail: [email protected]

Abstract

Classical Isometrodynamics is quantized in the Euclidean plus axial gauge.The quantization is then generalized to a broad class of gauges and the gener-ating functional for the Green functions of Quantum Isometrodynamics (QID)is derived. Feynman rules in covariant Euclidean gauges are determined andQID is shown to be renormalizable by power counting. Asymptotic states are

discussed and new quantum numbers related to the inner degrees of freedomintroduced. The one-loop effective action in a Euclidean background gaugeis formally calculated and shown to be finite and gauge-invariant after renor-malization and a consistent definition of the arising inner space momentumintegrals. Pure QID is shown to be asymptotically free for all dimensions ofinner space D whereas QID coupled to the Standard Model fields is notasymptotically free for D 7. Finally nilpotent BRST transformations forIsometrodynamics are derived along with the BRST symmetry of the theoryand a scetch of the general proof of renormalizability for QID is given.

1 IntroductionIn the search of a consistent new type of perturbatively renormalizable andunitary gauge field theory in four spacetime dimensions we have developedIsometrodynamics in [1]. This is the gauge theory of the group DIFFRD

of volume-preserving diffeomorphisms of a D-dimensional inner space(RD, g) endowed with a flat metric g.

We have formulated Classical Isometrodynamics in both the Lagrangianand the Hamiltonian frameworks and demonstrated that the theory canbe set up with a rigour similar to that achieved for classical Yang-Millsgauge theories even though the gauge group in our case is not compact

which brings along additional challenges.In the following we develop Quantum Isometrodynamics and show that

it can be formulated as a renormalizable and asymptotically free gaugefield theory in analogy to Yang-Mills theories of compact Lie groups [2, 3,4, 5].

The notations and conventions used are given in Appendix C.

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2 Quantization in the Euclidean plus Axial

Gauge

In this section we quantize Isometrodynamics in the Euclidean plus axialgauge deriving path integrals for Green functions with a manifestly in-variant gauge field weight and an invariant functional measure restrictedto the relevant gauge algebra diff RD.

Let us start with Hamiltonian Isometrodynamics in the Euclidean plusaxial gauge A3

M = 0 with Cartesian coordinates and Euclidean metric in inner space as developed in [1].

The D 2 independent canonical variables of the theory in this gaugeare Ai

M and their conjugates jN for i, j = 1, 2. They are subject to the

constraints

MAi M = 0Nj N = 0 (1)

which assure that the corresponding Ai = AiMM and j = j NN are

elements of the gauge algebra diff RD of infinitesimal volume-preservingcoordinate transformations of RD.

We next define an expression A0N non-local in the independent vari-

ables AiM, j

N - at this point just to keep the formulae below simple

A0M

1

3

2

1

2

i=1

DMi NiN, (2)

where and the covariant derivative

DMi N = i M

N + AiK K

MN NAi

M (3)

have been introduced in [1]. A0N fullfills

MA0M = 0 (4)

which is easily proven using the Eqn.(1).The Hamiltonian density HID of the theory is given by [1]

HID 2

i=1

iM DMi NA0

N +1

2

2i=1

iM i M

+2

4

2i,j=1

FijM Fij M (5)

+2

2

2i=1

3 AiM 3 Ai M

2

23 A0

M 3 A0M

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in terms of the A0M defined above and the expressions

FijM = iAj

M j AiM + Ai

N NAjM Aj

N NAiM. (6)

HID together with the Hamiltonian density HM =

n n 0n LMfor generic matter fields m with conjugates n is our starting point forpath integral quantization.

Green functions are defined as path integrals over AiM, j

N, m, nwith gauge and matter field measures

x,X,m

dm x,X,M,i=1,2

dAiM

i=1,2(MAi

M)

x,X,n

dn x,X,N,j=1,2

djN

j=1,2(Nj N) (7)

and weight

exp i

2

i=1

i M 0AiM HID +

n

n 0n HM

. (8)

Note that the -functions in the integration measures above ensure thatwe integrate over gauge fields and their conjugates belonging to the gaugealgebra diff RD only.

To turn these path integrals into Lorentz-invariant expressions we ap-ply the usual trick treating A0

M as a new independent variable whichwe can integrate over. The trick still works with the restricted measureEqn.(7). In fact, as the weight factor Eqn.(8) is at most quadratic in A0

M

we find that x,X,M

dA0M (MA0

M)

exp i

2

i=1

iM 0AiM HID

x,X,M

dA0M (MA0

M) (9)

exp i

2

2A0

N 32A0 N

exp i 12

2

i,j=1

DMi

KiK

1

32DjM

LjL + . . .

exp i

2

i=1

iM 0AiM HID

after a shift of integration variables A0M A0 M

132

1

2i=1 D

Mi Ki

K.This is - apart from a field-independent normalization factor - the gaugeweight factor Eqn.(8) with A0

M given by Eqn.(2) in terms of AiM, j

N.

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Next, as HID is quadratic in jN we can perform the corresponding

jN integrations for fixed Ai

M and A0M and find after a shift of integra-

tion variables jM j M F0j M

x,X,M,j=1,2

dj M j=1,2

(Mj M)

exp i

1

2

2i=1

iM i M +

2i=1

F0iM i M + . . .

x,X,M,j=1,2

djM

j=1,2(Mj M) (10)

exp i

1

2

2i=1

iM i M +

2

2

2i=1

F0iM F0i M + . . .

exp i 2

2

2

i=1 F0iM F0i M + . . . .

Above we have introduced

F0iM = 0Ai

M iA0M + A0

N NAiM Ai

N NA0M (11)

and used that F0iM is an element of the gauge algebra diff RD as is easily

verified.As a result Green functions are given as path integrals over Ai

M, A0N,

m - assuming that the integrations over n are Gaussian as well - withthe gauge field measure

x,X,MdA0 M (MA0 M) x,X,M,i=1,2dAi M i=1,2(MAi M) (12)

and gauge field weight

exp i

2

2

2i=1

F0iM F0i M

2

4

2i,j=1

FijM Fij M

2

2

2i=1

3 AiM 3 Ai M +

2

23 A0

M 3 A0M

. (13)

Introducing the variable A3M which vanishes identically in the Eu-

clidean plus axial gauge the integrals can finally be recast in a manifestlyLorentz-invariant fashion with gauge field measure

x,X,M,

dAM

(MA

M) (14)

and gauge field weight

(A3M) exp i

{LID + -terms} , (15)

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G[A, ] = exp i

{LID + LM + -terms} (21)

gauge invariant functionals of A,

x,X,n

dn = x,X,m

dm x,X,M,

dAM

(MA

M)

the integral J Eqn.(18) yields the Green functions of Isometrodynamicsin the Euclidean plus axial gauge as defined above. Here we have usedthe fact that

FR S[AM](x, X) = R S 3 (22)

is field-independent and Det F reduces to an overall normalization factorin the Euclidean plus axial gauge.

Next, let us check the gauge-invariance requirement Eqn.(19). Underlocal gauge transformations we have

AE M

= AM

+ E

M

+ AN

NE

M

E

N

NAM

,x,X,M,

dAE

M = Det

A

E M

A N

x,X,M,

dAM, (23)

(MAE M) = (MA

M).

Calculating

AE

M

A N=

MN + NEM EK K

MN

(24)

we find that the functional trace of the logarithm of the above Jacobianvanishes - yielding Det (. . .) = 1 in Eqn.(24). As a result the gauge fieldmeasure is gauge invariant and the condition Eqn.(19) is fulfilled.

Now we are in a position to freely change the gauge as path integrals ofthe form Eqn.(18) are actually independent of the gauge-fixing functionalfR[; x, X] and depend on the choice of the functional B [f] only throughan irrelevant constant. The proof of this crucial theorem is found e.g. in[2] - as all the steps in the proof hold true for Isometrodynamics as wellwe do not repeat them explicitly here.

As a result the generating functional for the Green functions of QIDin an arbitrary gauge and in the presence of matter fields is given by

Z[, J]

x,X,mdm

x,X,M,dA M (MA M) (25)

exp i

SID + SM +

2

J A +

m

m m + -terms

B [f[A, ]] Det F[A, ] ,

where we have introduced the external sources and J - transforming asa vector in inner space - for the matter and gauge fields respectively.

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whereSM OD SID + SGF + SGH (31)

is the modified FP gauge-fixed action for Isometrodynamics.

Eqn.(30) defines QID and is the starting point for the evaluation ofmatrix elements at the quantum level.

4 Perturbative Expansion, Feynman Rules

and Asymptotic States

In this section we derive the perturbative expansion of the generatingfunctional for the Green functions of pure QID and its Feynman rulesin Lorentz-covariant Euclidean gauges. We then use power counting todemonstrate the superficial renormalizability of QID. Finally we analyzethe asymptotic states of the theory and are led to introduce additionalquantum numbers related to the inner degrees of freedom of QID.

Working in Euclidean gauges with inner metric MN we use Eqn.(30)as the starting point for perturbation theory. Splitting the action

SM OD[A, , ] S0[A,

, ] + SIN T[A, , ] (32)

into the part S0 quadratic in the gauge and ghost fields and the interactionpart SIN T we can rewrite Eqn.(30) for pure Isometrodynamics as

Z[J, , ] = exp i SIN T

J,

,

Z0 [J, , ] , (33)

where

Z0 [J, , ]

x,X,M,

dAM

(MA

M)

x,X,R

dR (RR)

x,X,S

dS (S S) (34)

exp i

S0 + 2

(J A + + ) + -terms

is the the generating functional for Green functions of the non-interactingtheory and , are sources for the ghost fields. Note that for consistency

reasons all J, , have to be elements of the gauge algebra diff RD. Inparticular, it is crucial that the conserved gauge-field currents

JM = A N NF

M FN NA

M (35)

related to the global coordinate transformation invariance in inner spaceand generating the self coupling of the gauge fields are elements of thegauge algebra diff RD which is easily verified.

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To derive Feynman rules we have to specify the gauge and choose

fR[A] AR (36)

as the Lorentz-covariant gauge fixing function resulting in

FR S [A] =

R

S + AK K

RS SA

R

(37)

which is easily shown to be an endomorphism of diff RD as required.For the choice Eqn.(36) S0 is calculated to be

S0 = 2

2

D A

M D0, MN AN

2

D R DR0 S

S, (38)

where we have defined the non-interacting gauge and ghost field fluctua-tion operators by

D0, MN

2 +

1

1

MN

DR0 S 2 R S (39)

and the corresponding free propagators G0 through

D0, MR G0,

RN(x, y; X, Y) = MN D D(X Y)

4(x y)

DR0 M GM0 S(x, y; X, Y) = R S D D(X Y) 4(x y). (40)

The factors of ensure the scale invariance of the r.h.s under innerscale transformations. After some algebra we find the propagators for thegauge and ghost fields to be

G0,MN(x, y; X, Y) = D D(X Y) MN

d4k

(2)4ei k(xy)

1

k2 i

(1 )

kkk2

(41)

GR0 S(x, y; X, Y) = D D(X Y) R S

d4k(2)4

ei k(xy) 1k2 i

.

They are manifestly diagonal and local in inner space and invariant un-der local Euclidean transformations XM XM = AM(x)+OM N(x) X

N,OM N SO(D). The factors of naturally ensure that the integrationmeasure in inner K-space is dimensionless.

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Both the fluctuation operators and the propagators are endomorphismsof diff RD, i.e. iffM fullfills MfM = 0 so will D

0, MNf

N, G0,MNfN

and DR0 SfS, GR0 Sf

S as is easily verified. In other words the propaga-tors are the inverses of the fluctuation operators on the functional spacediff RD. As a consequence the -functions in the measure in Eqn.(34)will be automatically taken care of in the Gaussian integrals above.

Performing the Gaussian integrals over the gauge and ghost fields wefind

Z0 [J, , ] exp i

2

2

J M G

0,

MN J N

exp i 2

R GR0 S

S (42)

up to the functional determinants of the fluctuation operators Eqns.(39).

These field-independent normalization factors do not contribute to phys-ical amplitudes and can be discarded.Insertion of the result above into Eqn.(33) gives the unrenormalized

perturbation expansion of the generating functional of the Green functionsof QID which is plagued by the usual ultraviolet and infrared divergenciesof perturbative QFT. On top of these we will have to deal with potentiallydivergent integrals over inner space. We will show that they can beconsistently defined respecting the inner scale invariance of the classicaltheory.

Next, we give the momentum space Feynman rules which are easilyderived generalizing the usual approach by Fourier-transforming innerspace integrals as well.

The momentum space gauge field and ghost propagators are given by

G0,MN(k; K) =

1

k2 i

(1 )

kkk2

M N

GR0 S(k; K) =1

k2 i R S (43)

being unity in inner space. The inner degrees of freedom do notpropagate whereas the spacetime parts of the propagators equal the well-known Yang-Mills propagators.

The particle content is now easily read off - there is an uncountablyinfinite number of both massless gauge and unphysical ghost fields - thelatter to counter-balance the unphysical gauge field degrees of freedomarising in covariant gauges. Note that the positive-definiteness of the Eu-clidean metric M N with signature D is crucial to ensure unitarity of thetheory. An indefinite metric in inner space would make Isometrodyn-mics unviable as a physical theory.

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Next, we calculate the vertices starting with the tri-linear gauge fieldself-coupling

2 A M A M A N NA M (44)corresponding to a vertex with three vector boson lines. If these lines carryincoming spacetime momenta k1, k2, k3, inner momentum space coor-dinates K1, K2, K3 and gauge field indices M, N, L the contributionof such a vertex to a Feynman integral is

2 2

KL1 MN (k2 k2 )

+ KM2 NL (k3 k3 ) (45)

+ KN3 LM (k1 k1)

withk1 + k2 + k3 = 0, K1 + K2 + K3 = 0. (46)

The quadri-linear gauge field self-coupling term

2

2

A

N NAM A

N NAM

A R RA M (47)

corresponds to a vertex with four vector boson lines. If these lines carryincoming spacetime momenta k1, k2, k3, k4, inner momentum spacecoordinates K1, K2, K3, K4 and gauge field indices M, N, R, S thecontribution of such a vertex to a Feynman integral is

2

(KR1 KS2

MN KS2 KM3

N R + KM3 KN4

RS KR1 KN4

MS)

( )

+ (KS1 KR2

M N KS1 KN3

MR + KN3 KM4

RS KR2 KM4

NS)

( ) (48)

+ (KN1 KS3

M R KN1 KR4

M S + KM2 KR4

N S KM2 KS3

NR )

( )

withk1 + k2 + k3 + k4 = 0, K1 + K2 + K3 + K4 = 0. (49)

Finally, the gauge-ghost field coupling term

2 R

AS S

R S SAR

(50)

corresponds to a vertex with one outgoing and one incoming ghost lineas well as one vector boson line. If these lines carry incoming spacetimemomenta k1, k2, k3, inner momentum space coordinates K1, K2, K3 and

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field indices R, S, M the contribution of such a vertex to a Feynmanintegral becomes

2 (KM2 RS KS3

MR) k1 (51)

withk1 + k2 + k3 = 0, K1 + K2 + K3 = 0. (52)

In summmary, the above propagators and vertices allow us to evaluatethe Green functions of QID perturbatively. Note that for any Feynmangraph the analogue of the sums over Lie algebra structure constants inYang-Mills theories are integrals over inner momentum space variableswith the scale-invariant measure

dDK

(2)DD. (53)

As the vertices in such graphs contribute polynomials in the inner spacecoordinates KM to the integrand and as these inner degrees of freedomdo not propagate such integrals look badly divergent we will show in thenext section that they can be consistently defined respecting the innerscale invariance of the classical theory.

Turning to the spacetime integrals and renormalizability in the power-counting sense we note that the gauge and ghost fields have the samecanonical dimensions [A] = 1 and [] = [] = 1 relevant for powercounting as do their Yang-Mills counterparts.

The corresponding divergence indices 1 of the tri-linear gauge field

vertex, 2 of the quadri-linear gauge field vertex and 3 of the ghost-gaugefield vertex vanish

1 = 3 = b + d 4 = 3 + 1 4 = 0, 2 = 4 4 = 0, (54)

where b is the number of gauge field and ghost lines and d the numberof spacetime derivatives attached to the respective vertex. Accordinglythe superficial degree of divergence for any diagram with a total of Bexternal gauge field and ghost lines becomes

= 4 B (55)

which shows that only a finite number of combination of external lines willyield divergent integrals. As a result Isometrodynamics is renormalizableby power counting.

Let us finally consider the classification of asymptotic one-particlestates assuming they exist and are not confined which will be furtheranalyzed in the next section.

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To label the physical state-vectors we construct a basis of the one-particle Hilbert space of QID given by simultaneous eigenvectors of ob-servables commuting amongst themselves as well as with the Hamiltonianof the theory. In other words we look for a complete system of conserved,commuting observables for QID.

The specific difference of Isometrodynamics to a Yang-Mills theoryarises from the structure of the gauge group - all observables not related tothe gauge group remain the same and comprise the energy, the momentumand angular momentum three-vectors and other conserved internal degreesof freedom.

As Isometrodynamics and matter field Lagrangians minimally cou-pled to QID are translation and rotation invariant in inner space, wehave the corresponding conserved observables - the inner space mo-mentum operator KM and the inner angular momentum tensor. As

KM commutes with the already identified set of observables including theHamiltonian (consistent with the Coleman-Mandula theorem) its eigen-values KM become additional quantum numbers labelling physical states.In addition, as all matter fields transform as scalars and the gauge andghost fields as vectors under inner rotations inner spin becomes yetanother quantum number.

As a result we can find a basis of the one-particle Hilbert space

| k, ; KM, ; all other quantum numbers (56)

labeled by the momentum four-vector k, the spin , the inner space

momentum vector KM and the inner spin which is 0 for matterand 1 for the gauge and ghost fields of QID.

5 Effective Action, Renormalization at One-

Loop and Asymptotic Freedom

In this section starting from the formal perturbative expansion derived inthe last section we calculate the renormalized effective action at one loop.The crucial point is to note that spacetime and inner space integralsin the calculation of loop graphs completely decouple which allows us tofirst define the potentially divergent inner space integrals appropriately.Note that any consistent definition must respect the inner scale invari-ance of the classical action at the quantum level as this linearly realizedsymmetry is a symmetry of the quantum effective action as well [2]. Thisallows us second to deal in the usual way with the ultraviolet divergenciesrelated to the short distance behaviour in spacetime and demonstrate therenormalizability of QID at one loop.

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Technically we derive a formal expression for the one-loop effectiveaction of pure Isometrodynamics working in a covariant Euclidean back-ground field gauge. We then define the inner momentum integrals using as a cut-off and demonstrate the locality of the one-loop effective actionin inner space. To prove the renormalizability at one loop we calculatethe divergent contributions to the functional determinant of a generalfluctuation operator with differential operator-valued coefficients in fourspacetime dimensions. Finally we determine the one-loop counterterms,renormalize the one-loop effective action and calculate the -function ofboth pure Isometrodynamics and QID coupled to the Standard Modelfields.

5.1 Formal Expression

To derive a formal expression for the one-loop effective action of Isometro-dynamics we work in a covariant Euclidean background field gauge choos-ing

fR[A, B] BDR SA

S (57)

where

BDR S

RS + B

K K R

S SBR (58)

is the covariant derivative in the presence of a background field B we willfurther specify below.

To get the one-loop expression for the generating functional Eqn.(30)we have to expand the exponent around its stationary point up to second

order in the fluctuations. Starting with

SM OD[A, , ; B] =

2

4g2

D F

M F M

2

2g2

D BD

RMA

M BDR NA

N (59)

+ 2

D R FR

S [A, B] S,

where we have explicitly introduced a dimensionless gauge coupling g2

and where

FR S [A, B] = ES

BDR MAME |E=0

= BDR M D

MS (60)

is easily shown to be an endomorphism of diff RD as required, we get thefield equations in the presence of J and B

DN MFM +

1

BD

NR BD

R MA

M + g2 JN = 0

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BDR M D

MS

S = 0 (61)

BDS

M DMR R = 0.

They determine the stationary points AM = A

M[J, B], S = 0 and

R = 0 around which we expand. Setting the background field equal tothe stationary point

BM[J] =! A

M[J, B] (62)

determines B as a functional of J at least perturbatively.Next we calculate the second variation ofSM OD at the stationary points

2SM OD = 2

D AM DA, MN A

N

2 2

D R DR S

S, (63)

where we have absorbed the factors ofg in A M and calculated the gaugeand ghost field fluctuation operators to be

DA, MN DM

R DRN +

1

1

DM

R DRN

2 F RR MN + 2 N F

M (64)

DR S DRM DMS.

They are endomorphisms of diff RD, i.e. if fM fullfills MfM = 0 sowill DA, MNf

N and DR SfS, as is easily verified. Note that we had to

commute DRN

with DM

R to get the expression above for DA, MN

.Taking all together we finally get

Z1loop [J] =

x,X,M,

d AM

(MA

M)

x,X,R

d R (RR)

x,X,S

d S (S S)

exp i

SM OD[A, 0, 0; A] + 2

J A

exp

i

22

D A

M DA, M N AN (65)

i 2

D R DR S S + -terms

= exp i

SM OD[A, 0, 0; A] +

2 J A

Det 1/2 DA, Det D.

As the fluctuation operators are endomorphisms of diff RD the integralsin Eqn.(65) are Gaussian and can be performed resulting in the usual

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determinants. Indeed, endowed with the scalar product Eqn.(29), diff RD

becomes a Hilbert space with a complete orthonormal set of eigenvectorsfor each of the selfadjoint fluctuation operators above. These bases of theHilbert space take the -functionals automatically into account and theintegration over each eigenvector direction becomes Gaussian.

Defining next the generating functional for connected Green functions

W[J] i Ln Z[J] (66)

and the quantum effective action as its Legendre transform

[A]

J A + W (67)

in the usual way we find

1loop [A] = SM OD[A, 0, 0; A] + i2 Tr Ln DA, i Tr Ln D (68)

which is the formal expression for the one-loop effective action we werelooking for.

From now on we work with the specific choice = 1 and drop thesubscript to keep the calculations below as simple as possible.

5.2 Finiteness and Locality of Inner Space Inte-

grals

To get a well-defined quantum theory at the one-loop level we have toshow that the functional traces in Eqn.(68) above evaluated over the in-ner space can be appropriately defined, an issue which does not arise inYang-Mills theories of compact Lie groups due to the finite volume of theunderlying gauge groups.

To define Tr Ln DA and Tr Ln D and to demonstrate their locality ininner space note that both operators are of the form

D = 2

+ MIJ I

J

+NI I

+ C, (69)

where M MIJ|N,NM

I|N, CM

N are both matrices in inner space and matrix-

valued differential operators in Minkowski space. This form is very generaland can account for non-covariant Euclidean gauges such as the EuclideanLorentz gauge of Eqn.(57) as well, however, for = 1 the operator wouldtake an even more general form.

Properly normalizing and expanding the logarithm we obtain

Tr LnD

D0= Tr Ln D Tr Ln D0

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= Tr Ln

1 1

2

MIJ

IJ

+NI I

+ C

(70)

=

n()n

nTr

1

2

MIJ I

J

+NI I

+ C

n

=

n

()n

n(n),

where D0 is the operator for vanishing fields. Here we have defined theone-loop contribution with n vertex insertions

(n) Trx,X

1

2

MIJ

IJ

+NI I

+ Cn

=

dDX1 . . . . . .

dDXn

dDP1(2)D

. . . . . .

dDPn(2)D

Trx

X1 |

1

2

MI1J1

I1

J1 +NI1

I1 + C

| P1

...

Xn |

1

2

MInJn

InJn +NIn

In + C

| Pn

P1 | X2 . . . Pn | X1 (71)

=

dDX1 . . . . . .

dDXn

dDP1(2)D

. . . . . .

dDPn(2)D

Trx

1

2 MI1J1 iPI11 iPJ11 +NI1 iPI11 + CX1 ...

1

2

MInJn iP

Inn iP

Jnn +NIn iP

Inn + C

Xn

exp(iP1(X1 X2) + . . . + iPn(Xn X1))

which is manifestly invariant under local Euclidean transformations XM XM = AM(x) + OM N(x)X

N, OM N SO(D). Above we have inserted ncomplete systems of both X- and P-vectors

1 =

dDX | XX|, 1 =

d

D

P(2)D | PP|

and used X| P = exp(i P X) in Cartesian coordinates. Defining newvariables

K1 P1 Pn

K2 P2 P1, P2 = K2 + P1

17


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...... (72)

Kn1 Pn1 Pn2, Pn1 = Kn1 + . . . + K2 + P1

Kn Pn Pn1, Pn = Kn + . . . + K2 + P1

it becomes obvious that the definition above of the P1-integrals over poly-nomials in P1 requires care in order to avoid potential infinities related tothe non-compactness of the gauge group.

In generalization of our approach to defining the classical action ofIsometrodynamics we define such integrals using the cut-off introducedin [1]

(n)

dDX1 . . . . . .

dDXn

|P1|

dDP1(2)D

dDK2(2)D

. . . . . .

dDKn(2)D

Trx 1

2MI1J1 iPI1

1

iPJ11

+NI1 iPI1

1

+ CX1

... (73)

1

2

MInJn (iP

In1 + iK

In2 + . . . + iK

Inn )(iP

Jn1 + iK

Jn2 + . . . + iK

Jnn )

+ NIn (iPIn1 + iK

In2 + . . . + iK

Inn ) + C

Xn

exp(iX1(K2 + . . . + Kn) + iX2K2 + . . . + iXnKn) .

Next, using iKLj exp(in

l=2 XlKl) = L

j exp(in

l=2 XlKl) and partiallyintegrating we get

(n)

=

dDX1 . . . . . .

dDXn

|P1|

dDP1(2)D

dDK2(2)D

. . . . . .

dDKn(2)D

Trx

1

2

MI1J1 iP

I11 iP

J11 +NI1 iP

I11 + C

X1

... (74)

1

2

MInJn (iP

In1

In2 . . .

Inn )(iP

Jn1

Jn2 . . .

Jnn )

+ NIn (iPIn1

In2 . . .

Inn ) + CXn

exp(iK2(X2 X1) + . . . + iKn(Xn X1)) .

Above, the differential operators act to the left and ordering obviouslymatters. Integrating over Ki, Xj for i, j = 2, 3 . . .n yields the final ex-

pression for (n)

in this subsection

(n)

=

dDX1

|P1|

dDP1(2)D

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Trx

1

2

MI1J1 iP

I11 iP

J11 +NI1 iP

I11 + C

X1

... (75)

12

MInJn (iPIn1

In2 . . .

Inn )(iP

Jn1

Jn2 . . .

Jnn )

+ NIn (iPIn1

In2 . . .

Inn ) + C

Xn=Xn1=..=X1

.

The expression above for (n) is not only finite as an integral over inner

space, but also local in X1. Note that the regularized integrals over P1collapse into sums over products of -functions in inner space. Thesesums correspond to the sums over structure constants in the Yang-Millscase.

As in the case of the classical Lagrangian the contributions (n)

to theone-loop effective action for are related to the ones for a given by

(n)

(X, AM(X), . . .) =

(n) (X,A

M(X), . . .) (76)

respecting the scale invariance of the classical theory.At one loop the dependence of the theory on is again controlled

by the scale invariance of the classical theory. In other words up to oneloop theories for different are equivalent up to inner rescalings. Thissymmetry is not distroyed by the renormalization required for the diver-gent spacetime integrals with which we deal in the next subsection for the

simple fact that both types of integrals and how we properly define themcompletely decouple.

5.3 Divergence Structure of Spacetime Integrals

We turn to calculate the divergent contributions to the functional deter-minant of a general fluctuation operator with differential operator-valuedcoefficients in four spacetime dimensions in preparation of the one-looprenormalization in the next subsection.

To analyze the divergencies occurring in Tr Ln DA and Tr Ln D

note that both operators are of the form

D = 2

+ B

+ C, (77)

where B, C are both matrices in Minkowski space and matrix-valued dif-ferential operators in inner space. Again, this form is general enough tocope with non-covariant Euclidean gauges of the form Eqn.(57), however,the case = 1 is not included.

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Properly normalizing and expanding the logarithm we obtain

Tr LnD

D0= Tr Ln D Tr Ln D0

= Tr Ln

1 12

B + C

(78)

=

n

()n

nTr

1

2

B

+ C

n

=

n

()n

n

(n) ,

where D0 is the operator for vanishing fields. Here we have defined

(n)

Trx,X

1

2

B

+ C

n

=

d4x1 . . . . . .

d4xn

d4

p1(2)4 . . . . . .

d4

pn(2)4

TrX

x1 |

1

2

B1

11 + C

|p1

...

xn |

1

2

Bn

nn + C

|pn

p1 | x2 . . . pn | x1 (79)

= d4x1 . . . . . . d

4xn d4p1

(2)4

. . . . . . d4pn

(2)4

TrX

1

p21(i B1 p

11 + C)x1

...

1

p2n(i Bn p

nn + C)xn

exp(ip1(x1 x2) + . . . + ipn(xn x1)) ,

where we have inserted n complete systems of both x- and p-vectors

1 =

d4

x | xx |, 1 = d4p

(2)4 |pp |

and where x |p = exp(i p x). Note the occurrence of the propagatorsabove which is in marked difference to the local inner space integralsanalyzed in the last subsection.

A shift of variables

k1 p1 pn

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k2 p2 p1, p2 = k2 + p1...

... (80)

kn1 pn1 pn2, pn1 = kn1 + . . . + k2 + p1

kn pn pn1, pn = kn + . . . + k2 + p1

allows us to rewrite (n)

as

(n)

=

d4x1 . . . . . .

d4xn

d4p1

(2)4

d4k2

(2)4. . . . . .

d4kn(2)4

TrX

1

p21(i B1 p

11 + C)x1

... (81)

1

(p1 + k2 + . . . + kn)2 (i Bn (p

n

1 + k

n

2 + . . . + k

n

n ) + C)xn

exp(ix1(k2 + . . . + kn) + ix2k2 + . . . + ixnkn) .

Now it is easy to read off the degrees of divergence n for the p1-integralswhich are bound by n 4 n. Hence, only the

(n)

for n = 1, 2, 3, 4have a divergent contribution.

Isolating the divergent contributions which are local in x1 we findTr Ln

D

D0

div=

(1) div

1

2

(2) div

+1

3

(3) div

1

4

(4) div

= i

4

d4

x1 TrX

1

12

B

B

1

24 B B

+1

2 B C

1

2C2 (82)

+1

12 B B

B 1

12B

B B

1

4C B B

1

48B B B

B

1

96B B B B

.

Above, we have used the results from Appendix A for the

(n) div

forn = 1, 2, 3, 4 with = d 4 and 4 =1

82.

For fluctuations operators of the form

D = D D + E, D + A, (83)

where the gauge field A is a matrix-valued differential operator, we have

B = 2 A, C = A A A

+ E (84)

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and using the cyclicality property of the trace, which is easily demon-strated, Eqn.(82) further simplifies

Tr Ln DD0

div

= i4

d4 x1 Tr

X 1

12F F

+1

2E2. (85)

Above we have introduced the field strength operator

F [D, D] (86)

which belongs to the gauge field operator A.

5.4 One-Loop Renormalization

With the formulae Eqns.(85) and (86) which hold true in the presenceof general spacetime- and inner space-dependent fields we are now in

a position to analyze the one-loop renormalizability of Isometrodynamicsboth in the absence and presence of matter fields. Note that after prop-erly regularizing the inner space integrals we can safely interchange theorder of taking the traces over inner space versus spacetime variables.In this section we perform the functional trace over spacetime variablesfirst.

To analyze renormalizability we have to evaluate the divergent con-tributions to the one-loop effective action ,1loop [A] from Eqn.(68). Ashort calculation shows that the fluctuation operators Eqns.(64) take theform of Eqn.(83) above with

(A)M N = A KK M N NA M

(F)M

N = FKK

MN NF

M (87)

DA MN = (D)M

R (D)RN 2 (F)M NDR S = (D

)RM (D)MS .

As a result we get - after taking the trace over Minkowski indices - thedivergent contributions to the gauge field determinant

Tr LnDAD0

div= i

4

d4 x Tr

X

1

124 F F

+1

24 F F

= i 4

53

D

d4 x TrX

F F, (88)

and to the ghost determinantTr Ln

DD0

div= i

4

d4 x Tr

X

1

12F F

= i4

1

12D

d4 x TrX

F F. (89)

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Note that as for other gauge field theories it is the second term in Eqn.(88)which determines the sign of the gauge field contribution above which willin turn determine the sign of the -function of Isometrodynamics.

Taking all together we find

div,1loop [A] =

i

2

Tr Ln

DAD0

div i

Tr Ln

DD0

div

= 4

11

12D

d4x TrX

F F (90)

=4

D

D(D + 2)

11

12D 2

D F

M F M.

The one-loop divergence is proportional to the action of Isometrodynamicsand the theory is renormalizable at one loop. Note the formal similarity ofthe formula above with the analogous expression for Yang-Mills theories,especially the occurrence of the universal numerical factor 1112 .

As usual the divergent contribution div,1loop [A] can be absorbed in the

original action of Isometrodynamics through a redefinition of the gaugecoupling constant

gR = g

1 +

g2

42D

D(D + 2)

11

12D

1

+ O(g4)

(91)

where we have used 4 =1

82.

As a result the one-loop effective action after regularization of the

inner space integrals and renormalization is a perfectly well definedexpression.

The corresponding -function of Isometrodynamics at one loop be-comes

(g) = g3

42D

D(D + 2)

11

12D (92)

and the theory is asymptotically free.Note that does not get renormalized as we would expect from the

complete decoupling of inner and spacetime integrals and their treat-ments.

5.5 Inclusion of Standard Model Matter Fields

As discussed in [1] Isometrodynamics interacts with all fundamental fieldsappearing in a QFT such as the Standard Model (SM) of elementaryparticle physics through minimal coupling. For clarity we call all thesefundamental other scalar, spinor and (gauge) vector fields matter fields

23


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in the sequel. For a potential physical interpretation of Isometrodynam-ics it is hence crucial to extend the analysis of the asymptotic scalingbehaviour above to include the impact of these other fields on the renor-malized coupling and the -function.

To be specific let us do this analysis for the SM fields which we min-imally couple to Isometrodynamics by (1) allowing all SM fields to liveon M4 RD - adding the necessary additional inner degrees of free-dom - and by (2) replacing ordinary derivatives through covariant ones D = + A K K in all matter Lagrangians as usual.

In Appendix B we have derived the additional divergent contributionsdiv

,1loop [A] to the one-loop effective action contributing to the renor-malization of Isometrodynamcs.

To apply this to the SM let us recall its field content. The SM isbuilt on gauging SU(3) SU(2) U(1) which leaves us with 8 strongly,

3 weakly and 1 electromagnetically interacting gauge fields - 12 in total.These fields interact with 3 families of leptons and quarks, two of whichare structural replications of the first family consisting of the 15 chiralDirac fields for e, eL, eR, u

L, u

R, d

L, d

R, where = 1, 2, 3 indicates the

strongly interacting color degrees of freedom. Finally there is a Higgsdublett adding two scalar degrees of freedom.

In total we have

div,1loop [A]

div,1loop [A] + 12 G

div,1loop [A]

+ 45 Ddiv,1loop [A] + 2 S

div,1loop [A] (93)

= 4

D

D(D + 2)1

12

11D + 24 90 2

2

D F

M F M,

where 24 is the contribution of the SM gauge fields, 90 of the leptons andquarks and 2 of the Higgs respectively. This translates into the renormal-ized coupling

gR = g

1 +

g2

42D

D(D + 2)

1

12

11(D 6) 2

1

+ O(g4)

(94)

and the -function

(g) = g3

42D

D(D + 2)

1

12

11(D 6) 2

(95)

of Isometrodynamics coupled to the Standard Model fields.The combined theory is asymptotically free for 11(D 6) 2 > 0 or for

D 7. If the flip-side of asymptotic freedom is confinement the innerspace degrees of freedom and the gauge and matter fields associated

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with them are not expected to be directly observable - much as the gluonsand quarks in QCD.

Only for 2 D 6 do we expect the inner space degrees of freedomand the gauge and matter fields associated with them to be observableand asymptotic states to exist. In this case it makes sense to evaluate theclassical limit of Isometrodynamics.

Note that in the absence of the Higgs field the combined theory at oneloop for D = 6 is not renormalized at all.

6 BRST Symmetry and BRST Quantiza-

tion

In this section we introduce the nilpotent BRST transformations for Isome-

trodynamics and establish the BRST invariance of the gauge-fixed action.We define the physical states as equivalence classes of states in the kernelof the nilpotent BRST operator Q modulo the image of Q. Finally wediscuss the generalized BRST quantization of Isometrodynamics.

Let us start with the modified action SMOD from Eqn.(31) which maybe written as

SMOD = SID 2

2

fR f

R + 2

R R, (96)

where we have introduced the quantity

R FR S S. (97)

Next we reexpress

B[f] = exp

i

2

2

fR f

R

(98)

x,X,R

dhR (RhR) exp

i

2

2

hR h

R + i 2

hR fR

as a Gaussian integral and introduce the corresponding new modified ac-

tion

SNE W = SID + 2

R R + 2

hR f

R +2

2

hR h

R. (99)

Green functions are now given as path integrals over the fields A, , ,h, with weight exp i {SN EW + SM}.

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By construction the gauge-fixed modified action SN EW is not invariantunder gauge transformations. However, it is invariant under BRST trans-formations parametrized by an infinitesimal constant anticommutingwith ghost and fermionic fields. The BRST variations are given by

AM =

M + AKK

M KKAM

R = hR

S = KK

S (100)

hR = 0

= KK.

The transformations Eqns.(100) are nilpotent, i.e. if F is any functionalof A, ,,h, and we define sF by

F sF (101)

thensF = 0 or s(sF) = 0. (102)

The proof for the fields above is straightforward, but somewhat tedious.Here we just give a scetch ot the verification of s(sA

M) = 0

sAM =

KKM

+ K + A LLK LLA KKM A

KK

LLM

+

LLK

KAM (103)

+ KK

M + A

LLM LLA

M

= 0

using the chain-rule and the anticommutativity of with . As a resultwe have

s(sAM) = 0, s(sR) = 0, s(s

S) = 0

s(shR) = 0, s(s) = 0. (104)

The extension to products of polynomials in these fields follows then easily.To verify the BRST invariance of SNE W we note that the BRST trans-

formation acts on functionals of matter and gauge fields alone as a gaugetransformation with gauge parameter EM = M. Hence

SID = 0. (105)

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Next with the use of Eqn.(20) we determine the BRST transform of fR

fR =

fR

EM |E=0

M = R (106)

which yields

R R + hR f

R +

2hR h

R = s

R f

R +

2R h

R

. (107)

Hence we can rewriteSN EW = SID + s, (108)

where

2

R fR +

2R h

R

. (109)

It finally follows from the nilpotency of the BRST transformation

SN EW = 0. (110)

As for Yang-Mills theories Eqn.(108) shows that the physical content ofIsometrodynamics is contained in the kernel of the BRST transformationmodulo terms in its image.

Equivalent to this is the requirement that matrix elements betweenphysical states | , . . . are independent of the choice of the gauge-fixingfunctional . This implies the existence of a nilpotent BRST operator Qwith Q2 = 0. Physical states are then in the kernel of Q

Q | = 0, | Q = 0. (111)

Independent physical states are defined as the equivalence classes of statesin the kernel ofQ modulo the image ofQ.

Finally let us note that as for Yang-Mills theories [2] we can gener-alize the Faddeev-Popov-de Witt quantization procedure. In the generalcase one starts with an action given as the most general local functionalof A, , , h, with ghost number zero which is invariant under theBRST transformations Eqns.(100) and any other global symmetry of thetheory as well as with dimension less or equal to four so as to assurerenormalizability. Such actions are of the general form [2]

SNE W[A, , , h , ] = SID [] + s[A,

, , h , ] (112)

with s being a general functional respecting the restrictions above.S-matrix elements of physical states annihilated by the appropriate

BRST operator of the theory are then independent of . In addition, inthe Euclidean plus axial gauge the ghosts decouple in QID, hence theydecouple for any choice of and physical Isometrodynamics is ghost-free.

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7 Renormalizability to All Orders

In this section we scetch a proof of the renormalizability of Isometrody-namics to all orders.

A general proof of the renormalizability of Isometrodynamics, i.e. theexistence of a finite, well-defined perturbative effective action, has to com-prise the analysis of the divergence structure and the renormalizability ofspacetime integrals as for Yang-Mills theories and in addition the verifi-cation that inner space integrals can be properly defined respecting thescale invariance of the classical theory.

Turning to the first point we note that we should be able to employthe full machinery developed for the inductive renormalizability proof forYang-Mills gauge theories as the general structure of Quantum Isometro-dynamics formally is close to that of quantum Yang-Mills theories. Hence

we should be able to repeat all the steps in the renormalizability proof e.g.given in the Chapters 15 to 17 in [2] or in [6]. The only change arises fromthe slightly different form of the BRST transformations for Isometrody-namics as compared to Yang-Mills gauge theories requiring the adaptationof the analysis given in Section 17.2 of [2].

Turning to the second point our approach at the one-loop level hasbeen to (1) define the inner one-loop integrals using as a cut-off

dDP

(2)D integrand

|P|

dDP

(2)D integrand (113)

and (2) on the basis of this definition to demonstrate the validity of thescaling law

(1loop) (X, A

M(X), . . .) = (1loop) (X,A

M(X), . . .) (114)

ensuring the uniqueness of the theory up to inner rescalings.The same strategy should work for any number of loops. Again (1) we

define inner n-loop integrals by

dDP1(2)D

. . . dDPn(2)D

integrand (115)

|P1|

dDP1(2)D

. . . |Pn|

dDPn(2)D

integrand

arising in the calculation of the effective action and (2) on the basis of thisdefinition we should be able to demonstrate the validity of the scaling law

(nloop) (X, A

M(X), . . .) = (nloop)

(X,AM(X), . . .) (116)

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noting that the inner scale invariance is a linearly realized symmetry ofIsometrodynamics and hence a symmetry of the quantum effective action[2]. This ensures the uniqueness of the theory up to inner rescalings atn loops.

The locality of the theory in inner space for any number of loopsfollows from the non-propagation of inner degrees of freedom which canbe most easily read off the propagators in Eqns.(41).

This completes the scetch of a general proof of the renormalizabilityand the essential uniqueness of Quantum Isometrodynamics.

8 Conclusions

In this paper we have developed Isometrodynamics at the quantum level.

Important aspects of the quantization have been dealt with in veryclose analogy to the quantization of Yang-Mills gauge theories - in partic-ular all the aspects related to (1) dealing with the pure gauge degrees offreedom, to (2) developing a perturbation theory and to spacetime-relateddivergencies of Feynman integrals and to (3) the asymptotic behaviour ofthe theory. The generalization of all these aspects to the infinitely manyinner degrees of freedom of QID have posed no fundamentally new prob-lems.

However, there were new challenges to be adressed - all related to thegauge group DIFFRD. First, the gauge field and ghost variables A

M

and R, S which naturally emerge from the gauging program had to be

subject to restrictions which ensure that they live in the appropriate gaugealgebra diff RD. These restrictions are not constraints in the usual senseand had to be implemented in the theory in a consistent way, i.e. throughconstraints in the gauge field and ghost functional measures for the pathintegrals for the theorys Green functions. Second, the gauge group is notcompact and to avoid divergencies related to the infinite group volume wehad to properly deal with the sums over inner degrees of freedom. Theybecome integrals over inner coordinates in Isometrodynamics and had tobe defined through a regularization procedure respecting the inner scaleinvariance of the theory at the quantum level - making QID unique up to

inner rescalings. Third, the inner degrees of freedom, in particular theinner global translation invariance of Isometrodynamics and the relatedconserved inner momenta have brought along new quantum numberswhich will eventually require interpretation.

As an overall result Isometrodynamics viewed as a generalization ofnon-Abelian gauge theories of compact Lie groups seems to stand on asolid basis as a new type of renormalizable gauge theory.

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But there is an obvious question here. Can QID be used to describefundamental interactions in Nature at both the classical and quantum levelwith gravity being a central candidate? Or is QID just a mathematicalgeneralization of a framework which is limited to successfully describe thestrong, weak and electromagnetic interactions in Nature?

To shed some light on this question let us point out that by its verydefinition Isometrodynamics fulfills important requirements towards anytheory of gravity such as universality, i.e. the universal coupling of grav-ity to all fundamental particles and fields or such as the existence of asensible limit for the case of gravity being turned off, i.e. the laws ofphysics reducing to the Standard Model of elementary particle physics ora generalization thereof.

We will separately discuss the potential of Isometrodynamics to be atheory of gravity.

A Dimensional Regularization of Divergent

One-Loop Integrals

In this Appendix we calculate the divergent contributions to the one-loop functional determinant Eqn.(78) in four spacetime dimensions. |divindicates that we will retain only divergent terms and discard all finitecontributions in a calculation.

A.1 The contribution (1)div linear in the fields

(1) div =

d4x1

d4p1

(2)4TrX

1

p21

i B1(x1)p

11 + C(x1)

|div

(117)

which identically vanishes using dimensional regularization,

1

(2)4

d4p

1

p2

i

(2)d

ddp

1

p2= 0 (118)

For the evaluation we have Wick-rotated the integral p0 ip4 and ana-lytically continued it to d = 4 dimensions as we will do for all theintegrals below.

A.2 The contribution (2)div

quadratic in the fields

(2) div =

d4x1

d4x2

d4p1

(2)4

d4k2

(2)4

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TrX

1

p21

i B1(x1)p

11 + C(x1)

1

(p1 + k2)2 i B2(x2) p

21 + k

22 + C(x2)

|div

exp(ix1k2 + ix2k2)

=

d4x1

d4x2

d4p1

(2)4

d4k2

(2)4(119)

TrX

p1 p

1

p21(p1 + k2)2

i B(x1) i B(x2)

+p1

p21(p1 + k2)2

i B(x1) i B(x2) k

2

+ i B(x1) C(x2) + C(x1) i B(x2)

+ 1

p21(p1 + k2)2

C(x1) i B(x2) k2 + C(x1) C(x2)

|div

exp(ix1k2 + ix2k2) .

We evaluate

1

(2)4

d4p

1

p2(p + k2)2 |div i

4

(120)

1

(2)4

d4p

p

p2(p + k2)2 |div i

k22

4

(121)

1(2)4

d4p p

p

p2(p + k2)2 |div

i3

k2 k2 i12

k22

4

(122)

and obtain

(2) div = i

4

d4 x1 Tr

X

1

6 B(x1)

B(x1)

+1

12 B(x1) B

(x1) (123)

B(x1) C(x1) C2(x1)

after rewriting k2 exp(ix2k2) = i

2 exp(ix2k2), partially integrating

2 ,integrating out k2, x2 and using the cyclicality of the trace which is easilyshown to hold also true in the case ofB, C being matrix-valued differentialoperators acting on inner space coordinates.

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cubic in the fields

(3) div

= d4x1 d

4x2 d4x3

d4p1

(2)

4d4k2

(2)

4d4k3

(2)

4

TrX

1

p21

i B1(x1)p

11 + C(x1)

1

(p1 + k2)2

i B2(x2)

p21 + k

22

+ C(x2)

1

(p1 + k2 + k3)2

i B3(x3)

p31 + k

32 + k

33

+ C(x3)

|div

exp(ix1(k2 + k3) + ix2k2 + ix3k3)

= d4x1 d

4x2 d4x3

d4p1(2)4

d4k2(2)4

d4k3(2)4

(124)

TrX

p1 p

1 p

1

p21(p1 + k2)2(p1 + k2 + k3)2

i B(x1) i B(x2) i B(x3)

+p1 p

1

p21(p1 + k2)2(p1 + k2 + k3)2

i B(x1) i B(x2) C(x3)

+ i B(x1) C(x2) i B(x3) + i C(x1) B(x2) i B(x3)

+ i B(x1) i B(x2)i B(x3)(k2 + k

3 )

+ i B(x1)i B(x2)k2 i B(x3)

|div

exp(ix1(k2 + k3) + ix2k2 + ix3k3) .

After evaluating

1

(2)4

d4p

pp

p2(p + k2)2(p + k2 + k3)2 |div

i

4

4

(125)

1

(2)4

d4p

ppp

p2(p + k2)2(p + k2 + k3)2 |div(126)

i

12

(2k2 + k

3) +

(2k2 + k3 ) +

(2k2 + k3) 4

we obtain

(3) div = i 4

d4 x1 TrX

14 B(x1) B(x1) B(x1) (127)

1

4B(x1)

B(x1) B(x1) 3

4C(x1) B

(x1) B(x1)

.

Here we have rewritten kj exp(ixj kj) = i

j exp(ixj kj), partially inte-grated

j , integrated out kj, xj for both j = 2, 3 and used the cyclicality

of the trace.

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quartic in the fields

(4) div = d4x1 d4x2 d4x3 d4x4 d

4p1

(2)4 d

4k2

(2)4 d

4k3

(2)4 d

4k4

(2)4

TrX

1

p21

i B1(x1)p

11 + C(x1)

1

(p1 + k2)2

i B2(x2)

p21 + k

22

+ C(x2)

1

(p1 + k2 + k3)2

i B3(x3)

p31 + k

32 + k

33

+ C(x3)

1

(p1 + k2 + k3 + k4)2

i B4(x4) (128)

p

41 + k

42 + k

43 + k

44

+ C(x4)

|div

exp(ix1(k2 + k3 + k4) + ix2k2 + ix3k3 + ix4k4)

=

d4x1

d4x2

d4x3

d4x4

d4p1

(2)4

d4k2

(2)4

d4k3

(2)4

d4k4

(2)4

TrX

p1 p

1 p

1p

1

p21(p1 + k2)2(p1 + k2 + k3)2(p1 + k2 + k3 + k4)2

i B(x1) i B(x2) i B(x3) i B(x4)

|div

exp(ix1(k2 + k3 + k4) + ix2k2 + ix3k3 + ix4k4) .

With the use of

1

(2)4

d4p

pppp

p2(p + k2)2(p + k2 + k3)2(p + k2 + k3 + k4)2 |div

i

24

+ +

4

(129)

we obtain

(4) div

= i4

d4 x1 TrX

1

12

B(x1) B(x1) B(x1) B(x1)

+1

24B(x1) B

(x1) B(x1) B(x1)

(130)

after integrating out kj, xj for all j = 2, 3, 4 and using the cyclicality ofthe trace.

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B Matter Contributions to Divergent Part

of One-Loop Effective Action of Isometrody-

namics

In this Appendix we calculate the divergent vacuum contribution of agauge vector field, a Dirac spinor and a complex scalar doublet to theone-loop effective action of Isometrodynamics.

B.1 Gauge field contribution Gdiv

,1loop [A]

The vacuum amplitude of a Yang-Mills gauge field B with gauge alge-

bra indices ,,.. = 1,.., dim A minimally coupled to Isometrodynamics,where dim A is the dimension of the gauge algebra, is given by

ZG[A]

x,X,,

dB

x,X,

d

x,X,

d

exp i {SMOD + -terms} , (131)

where D [B] =

+ C

B is the covariant derivative in the pres-

ence of a gauge field B, C the structure constants of the gauge algebraand ,

the ghost fields corresponding to the gauge-fixed action SM OD

SMOD SY M + SGF + SGH

SY M 1

4

G

G

(132)

SGF 12

D

[C]B

D[C]B

SGH

F

[B, C] .

C appearing in the gauge-fixing and ghost terms is a background gauge

field. Above we have minimally coupled the Yang-Mills field to Isometro-dynamics replacing ordinary through covariant derivatives D = + A

K K yielding

D [B] D [B] = D

+ C

B

, (133)

and introduced the field strength and the ghost fluctuation operator

G = DB

DB + C B

B,

F [B, C] = D[C] D

[B]. (134)

The bars over derivatives etc. indicate minimal coupling to Isometrody-namics.

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Expanding SM OD around its stationary points B = C

= = = 0 in the absence of source terms and performing the Gaussian inte-gral gives

ZG,1loop[A] =

x,X,,

dB

x,X,

d

x,X,

d

exp

i

2

B

DB, B (135)

D

= Det 1/2 DB, Det D,

where

D

B,

D

D +

1

1

D

D

F

D D D

. (136)

Taking everything together and evaluating the divergent contributionto the one-loop effective action with the use of Eqns.(136), (85) and (86)for = 1 yields for each independent gauge field and associated ghost

Gdiv,1loop [A] =

4

D

D(D + 2)

1

62

D F

M F M, (137)

where we have discarded the factor dim A which accounts for the number

of independent gauge fields. Note that such a term will reinforce asymp-totic freedom. Note in addition that this formula also holds in the Abeliancase where the ghost contribution in the presence of A

M does not reduceto a field-independent determinant.

B.2 Dirac spinor contribution Ddiv

,1loop [A]

The vacuum amplitude of a Dirac field minimally coupled to Isometrody-namics is given by

ZD[A] x,Xd x,Xd exp i {SD + -terms} , (138)where is a Dirac spinor and

SD

(D/ + m) (139)

is the spinor action coupled to a Yang-Mills field through the covariantderivative D[B] = i tB

. Here t is the generator of the gaugealgebra in the fermion space.

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Again we have minimally coupled the Dirac field to Isometrodynamicsreplacing ordinary through covariant derivatives D = +A KKyielding

D[B] D[B] = D i tB. (140)

Expanding SD around its stationary points = = B = 0 in the

absence of external sources and performing the Grassmann integral gives

ZD,1loop[A] =

x,X

d

x,X

d exp

i

D

= Det 1/2 D2, (141)

where

D2 = D/2 = D D

1

2F. (142)

Taking everything together and evaluating the divergent contributionto the one-loop effective action with the use of Eqns.(142), (85) and (86)yields for each independent Dirac spinor

Ddiv,1loop [A] =

4

D

D(D + 2)

1

32

D F

M F M. (143)

Note that this will work against asymptotic freedom. Note in additionthat a chiral Dirac fields contributes just half of the value above.

B.3 Scalar doublet contribution Sdiv

,1loop [A]

The vacuum amplitude of a complex scalar doublet minimally coupled toIsometrodynamics is given by

ZS[A]

x,X

d

x,X

d exp i {SS + -terms} , (144)

where is a complex scalar doublet and

SS

(D) (D) + V(

)

(145)

is the doublet coupled to the SU(2) U(1) gauge bosons of the electro-weak interaction through the covariant derivative D[B] = i B t i B y.

Again we have minimally coupled the scalar to Isometrodynamics re-placing ordinary through covariant derivatives D = + A K Kyielding

D[B] D[B] = D i B

t

i B y. (146)

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Expanding SS around one of its stationary points B

= B = 0 and = const. and performing the Gaussian integral gives

ZS,1loop[A] = x,Xd

x,Xd expi D

= Det 1 D (147)

where

D = D D +

V( )

. (148)

Taking everything together and evaluating the divergent contributionto the one-loop effective action with the use of Eqns.(148), (85) and (86)yields for a complex scalar doublet

Sdiv,1loop [A] =

4

D

D(D + 2)

1

62

D F

M F M, (149)

which holds independent of whether the Higgs mechanism is in place ornot and will work against asymptotic freedom. Note that a single complexscalar field contributes just half of the value above.

C Notations and Conventions

Generally, small letters denote spacetime coordinates and parameters,capital letters coordinates and parameters in inner space.

Specifically, (M4 , ) denotes 4-dimensional Minkowski spacetime with

the Cartesian coordinates x

, y

, z

, . . . and the spacetime metric =diag(1, 1, 1, 1). The small Greek indices ,, , . . . from the middle ofthe Greek alphabet run over 0, 1, 2, 3. They are raised and lowered with, i.e. x = x

etc. and transform covariantly w.r.t. the Lorentz groupSO(1, 3). Partial differentiation w.r.t to x is denoted by

x

. SmallLatin indices i , j , k , . . . generally run over the three spatial coordinates1, 2, 3 [?].

(RD, g) denotes a D-dimensional real vector space with coordinatesXL, YM, ZN, . . . and the flat metric gMN with signature D. The metrictransforms as a contravariant tensor of Rank 2 w.r.t. DIFFRD. Be-cause Riem(g) = 0 we can always choose global Cartesian coordinatesand the Euclidean metric = diag(1, 1, . . . , 1). The capital Latin indicesL , M , N , . . . from the middle of the Latin alphabet run over 1, 2, . . . ,D.They are raised and lowered with g, i.e. XM = gMNX

N etc. and trans-form as vector indices w.r.t. DIFFRD. Partial differentiation w.r.t toXM is denoted by M

XM .

The same lower and upper indices are summed unless indicated other-wise.

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Acknowledgments

This paper is dedicated to my daughters Alina and Sarah and to my wifeFrancoise who have helped me to keep the right perspectives on this work

and who have patiently carried me through the sometimes emotional upsand downs on the stony way to Isometrodynamics.

References[1] C. Wiesendanger, Classical Isometrodynamics

[2] S. Weinberg, The Quantum Theory of Fields II (Cambridge Univer-sity Press, Cambridge, 1996).

[3] S. Pokorski, Gauge Field Theories (Cambridge University Press,Cambridge, 1987).

[4] C. Itzykson, J-B. Zuber, Quantum Field Theory (McGraw-Hill, Sin-gapore, 1985).

[5] T-P. Cheng, L-F. Li, Gauge Theory of Elementary Particle Physics(Oxford University Press, Oxford, 1984).

[6] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena (Ox-ford University Press, Oxford, 1993).

c. wiesendanger- quantum isometrodynamics

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