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204 J. Opt. Soc. Am. A/Vol. 24, No. 1 /January 2007 T. Tudor and A. Gheondea

Pauli algebraic forms of normaland nonnormal operators

Tiberiu Tudor

Faculty of Physics, University of Bucharest, P.O. Box MG-11, 077125 Bucharest-Magurele, Romania

Aurelian Gheondea

Department of Mathematics, Bilkent University, 06800 Bilkent, Ankara, Turkey, and Institute of Mathematicsof Romanian Academy, P.O. Box. 1-764, 014700 Bucharest, Romania

Received March 17, 2006; revised June 18, 2006; accepted July 3, 2006;posted July 27, 2006 (Doc. ID 69093); published December 13, 2006

A unified treatment of the Pauli algebraic forms of the linear operators defined on a unitary linear space of twodimensions over the field of complex numbers C1 is given. The Pauli expansions of the normal and nonnormaloperators, unitary and Hermitian operators, orthogonal projectors, and symmetries are deduced in this frame.A geometrical interpretation of these Pauli algebraical results is given. With each operator, one can associatea generally complex vector, its Pauli axis. This is a natural generalization of the well-known Poincaré axis ofsome normal operators. A geometric criterion of distinction between the normal and nonnormal operators bymeans of this vector is established. The results are exemplified by the Pauli representations of the normal andnonnormal operators corresponding to some widespread composite polarization devices. © 2006 Optical Soci-ety of America

OCIS codes: 260.5430, 000.3860.

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. INTRODUCTIONn the Pauli algebraic theory of the polarization device op-rators, it is common knowledge that an ideal general po-arizer has the representation1–3

P�n� =1

2��0 + n · ��, �1�

hereas the representation of a general retarder is1–4:

R�n,�� = ei��/2�n·� = �0 cos�

2+ in · � sin

�

2. �2�

ere n is a three-dimensional real unit vector and ��1 ,�2 ,�3�, with �i being the Pauli matrices:

�0 = �1 0

0 1�, �1 = �0 1

1 0�, �2 = �0 − i

i 0 � ,

�3 = �1 0

0 − 1� . �3�

Many concrete problems in the field of light polariza-ion are solved by a Pauli algebraic approach startingith these formulas.1–8 But, more generally, the Pauli al-ebra is a powerful and widespread tool in handling theroblems of any two-states quantum systems.9–11 Theauli algebraic approach to these systems is a fundamen-al one, because it addresses directly the actual topologyf their Hilbert state space, which is isomorphic with thatf the Poincaré–Bloch sphere.12,13

1084-7529/06/010204-7/$15.00 © 2

From a mathematical viewpoint, formulas (1) and (2)epresent the Pauli algebraic expansions of some veryarticular kinds of normal operators defined on a unitaryinear space of two dimensions over the field of complexumbers C1, namely, orthogonal projectors of rank onend unitary operators, respectively.In recent times the nonnormal operators have come to

ight, in matrix forms, in several areas of physics,14–16 inonnection with a large variety of problems: mode degen-racies for unstable lasers,17 light propagation in biaxialbsorbing and chiral crystals,18 diffraction of atomiceams by “crystals of light,”19 non-Hermitian “nonphys-cs” of a pile of plates,1 phase transition in open quantumystems,20 level or resonance crossings and anti-rossings,21–24 etc.

Referring to the field of light polarization, the operatorsf all the basic (canonical) polarization devices (homoge-eous polarizers and retarders) are normal operators.25

n the other hand, the operators of the compositemultilayer) polarization devices may be normal as wells nonnormal,26,27 The nonnormal (non-Hermitian) polar-zers play an important role in connection with the theoryf the generalized quantum measurement.28–31

We have to stress that the Pauli algebraic expansionsf the orthogonal projectors and unitary operators [Eqs.1) and (2)] are two isolated formulas that were estab-ished inductively, on experimental ground, in the field ofolarization theory.32 To the best of our knowledge, noystematic theory of the Pauli algebraic forms of the vari-us kinds of operators was elaborated upon until now.

Bearing in mind the above-mentioned enlargement of

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T. Tudor and A. Gheondea Vol. 24, No. 1 /January 2007 /J. Opt. Soc. Am. A 205

he class of operators implied in solving new problems ofhe two-state quantum systems, it seems that it is theight time to give a coherent theory of the Pauli forms ofhe operators on two-dimensional unitary space.

The aim of this paper is to give a unified Pauli algebraicreatment of the linear operators defined on a unitarypace of two dimensions over the class of complex num-ers C1. In this framework we shall fill the gap betweenormulas (1) and (2) by establishing the hierarchy of Paulixpansions of the operators of this class, and by integrat-ng deductively Eqs. (1) and (2) of the orthogonal projec-ors and unitary operators in this hierarchy.

A general result we obtain is that the Pauli algebra pro-ides a remarkable criterion of distinction between theormal and nonnormal operators: in this approach, toach operator on a complex vector space of two dimen-ions there corresponds a vector in C3, which we shall callhe Pauli axis of the operator. The operator is normal ifnd only if its Pauli axis is a real vector or is reducible toreal vector by a phase shift. The Pauli axes of nonnor-al operators are irreducible complex vectors.Finally we exemplify our general results by deducing

he Pauli expansions of the operators of some widespreadrthogonal and nonorthogonal composite polarization de-ices.

. NORMAL OPERATORSe shall consider a linear operator A�L�V�, where V is a

inear space of two dimensions over the field of complexumbers C1. L�V� is identified with the algebra of 2�2atrices with complex entries. It is well known that the �atrices constitute a basis in the vector space of these op-

rators, so that any such operator may be expressed inhe form

A = a0�0 + a · �, �4�

here a0 is a generally complex scalar, and a is a gener-lly complex three-dimensional vector. For reasons thatill become evident later on we shall denominate the vec-

or a the axis of the operator. The coefficients a0 ,a1 ,a2,nd a3, the components of a, are known in the particularase of Hermitian operators, in polarization optics, underhe name of Stokes coefficients (parameters), in whichase they are real. We will extend this denomination forny (2�2 matrix) operator.If we label by A† the adjoint of A:

A† = a0*�0 + a* · �, �5�

he condition of normality of A is

AA† = A†A. �6�

By using Dirac’s equation concerning the Pauli expan-ion of the product of two operators A [Eq. (4)] and B,

B = b0� + b · �, �7�

amely (e.g., Ref. 2):

AB = �a0b0 + a · b��0 + �b0a + a0b� · � + i�a � b� · �.

�8�

quation (6) takes on the form

�a0a0* + a · a*��0 + �a0

*a + a0a*� · � + i�a � a*� · �

= �a0*a0 + a* · a��0 + �a0a* + a0

*a� · � + i�a* � a� · �.

�9�

earing in mind the anticommutativity of the outer prod-ct of two vectors, Eq. (9) leads to

a � a* = 0. �10�

As a consequence, the two complex-conjugate vectors and a* must be collinear, i.e.,

a* = �a, �11�

here � is a complex number of modulus 1.Condition (11) means that, apart from a complex factor,

he Pauli axis of a normal operator reduces to a real vec-or:

a = ei�r. �12�

Hence the Pauli expansion of a normal operator is

A = ei�0�a0��0 + ei�r · �, �13�

here r is some real vector, �0 is a real scalar modulo 2�,nd � is a real scalar modulo �.Obviously, the adjoint of operator A is

A† = e−i�0�a0��0 + e−i�r · �. �14�

Finally we shall note that for a normal operator

AA† = A†A = ��a0�2 + �r�2��0 + 2�a0�r · � cos�� − �0�,

�15�

here we have made use of Eqs. (13), (14), and (10).Let us deduce now from the general Pauli expansion ofnormal operator [Eq. (13)] some particular forms, corre-

ponding to the most widespread kinds of normal opera-ors.

. Unitary Operatorsor a unitary operator we have

UU† = I � �0, �16�

nd with Eq. (15) we get two simultaneous equations:

�a0�r cos�� − �0� = 0, �17�

�a0�2 + �r�2 = 1. �18�

If a0�0, from Eq. (17) we get:

� − �0 = �/2 modulo �, �19�

nd, on the other hand, Eq. (18) may be fulfilled if we put:

�a0� = cos �/2, r = n sin �/2, �20�

ith n a real unit vector.Coming back to Eq. (13) with Eqs. (19) and (20) we ob-

ain the most general Pauli algebraic form of the unitaryperators:

U = ei�0�� cos �/2 + in · � sin �/2� = ei�0ei��/2�n·�. �21�
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From Eq. (18), if a0=0, then r=n, a unit vector, and thearticular case of Eq. (21) is obtained, namely a symme-ry (see Subsection 2.D). Equation (21) gives Eq. (2) up tophase factor.In polarization optics, such an operator corresponds torotation of angle � of the polarization state on the

oincaré sphere, around an axis defined by the unit vec-or n, which can be obviously named the Poincaré axis ofhe operator. For a unitary operator its axis is a real unitector n, figurable in the real three-dimensional space. Its natural to extend the term of axis of the operator (theauli axis of the operator) for the generally complex vec-or a in Eq. (4).

In these terms, the remarkable difference between theormal and the nonnormal operators is that the axes ofhe normal operators are, apart from a phase factor, realectors, whereas the axes of the nonnormal operators areomplex vectors.

If we refer specifically to the optical polarization de-ices, the operators of the retarders are unitary operators,hey pertain to the SU�2� group, which is isomorphic with�3�. Therefore an intuitive geometrical representation in3 for them may be given: a real axis, its Poincaré axis,

an be associated to each retarder, and the action of theetarder on the incident light can be represented as a ro-ation around this axis in R3, more precisely on theoincaré sphere. Generally, the operators of the (nondepo-

arizing) optical devices pertain to the six-parameterL(2,c) group. For grasping a geometrical insight into thection of such an operator, our intuition has to transcendn a six-dimensional real space, or equivalently in a three-imensional complex space.

. Hermitian Operatorsor a Hermitian operator,

H = H†, �22�

ith Eqs. (13) and (14) and bearing in mind that r is aeal vector, we get 2�0=0, modulo 2�, i.e.,

�0 = 0,� �modulo 2��, �23�

nd 2�=0 modulo �, i.e.,

� = 0 �modulo ��. �24�

ence for a Hermitian operator all the Stokes coefficientsre real:

H = ± �a0��0 + r · �. �25�

. Orthogonal Projectorsspecial kind of Hermitian operators are the orthogonal

rojectors. A projector is idempotent:

P2 = P. �26�

With P given by Eq. (25) and making use of Dirac’squation

�a · ��b · �� = �a · b� + i�a � b� · �, �27�

or a=b=r, Eq. (26) becomes

��a �2 + �r�2�� + 2�a �r · � = ± �a �� + r · �. �28�
0 0 0 0 0
By identifying between the two members of Eq. (28), weet

�a0�2 + �r�2 = ± �a0�, �29�

2�a0�r = r. �30�

Let us now analyze the various situations coming fromhese two simultaneous equations. If we take the minusign in Eq. (29) it follows that

�a0� = 0, �r� = 0, �31�

.e., from Eq. (25):

P = O, �32�

here we have labeled O the null operator.If we want to have P�O, we should take in Eq. (29)

nd, consequently in Eq. (25) the plus sign. Further onhere are two situations. If r=0, from Eq. (29) we geta0 � =1, and with Eq. (25):

P = �0 � 1. �33�

inally if r�0, Eq. (30) leads to �a0 � = 12, hence

P =1

2��0 + n · ��, �34�

ith n a real unit vector, in accordance with Eq. (1).If we refer again to the polarization devices, Eq. (34) is

he operator of an orthogonal polarizer whose Poincaréxis is n (evidently, such an operator does not determine aotation of the state on the Poincaré sphere around itsxis, but a spherical projection on its axis).

. Symmetriessymmetry is a unitary and self-adjoint operator. Note

hat all the four Pauli matrices are symmetries. Also, S issymmetry if and only if

S = 2P − l, �35�

here P is an orthogonal projection. Thus, if we excludehe extremal cases P=0 and P= I from Eq. (34) we get that

is a symmetry if and only if

S = n · �, �36�

here n is a real unit vector.

. PAULI EXPANSION OF SOME NORMALND NONNORMAL DEVICE OPERATORS

N POLARIZATION OPTICShe Pauli algebraic forms of various normal device opera-

ors, either Hermitian (corresponding to orthogonal polar-zers) or unitary (corresponding to retarders), are largelysed in polarization optics.1–8,33,34 In particular, all theomogeneous canonical optical devices are of the orthogo-al kind (orthogonal eigenvectors).25

But it is well known18,35,36 that for those media that ex-ibit simultaneously optical activity, linear birefringence,nd linear dichroism, the eigenstates are no longer or-hogonal. On the other hand, some of the inhomogeneous,omposite devices can be nonorthogonal (skew eigenvec-

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nto consideration28 in connection with the subject of theeneral quantum measurement.29–31 In spite of all this,he investigations concerning the nonnormal operatorsemain somewhat peripheral in the polarization theory.n particular no analysis of the Pauli algebraic form of theperators of nonorthogonal polarization devices has beenone until now. Therefore, we shall deduce here the Paulixpansions of the (nonnormal, non-Hermitian) operatorsf some widespread nonorthogonal (skew) polarizers.

. Nonorthogonal Deviceshe simplest example of a nonnormal device operator inolarization optics is that of a non-Hermitian linear po-arizer obtained by sandwiching together two normal lin-ar polarizers at different azimuths ��0,� /2�:

P = P�P��P�Px�. �37�

The Poincaré axes of the two Hermitian polarizers P�Px�nd P�P� � being

nx�1,0,0�,

n��cos 2�,sin 2�, 0�, �38�

heir Pauli expansions are, respectively,

P�P�� =1

2��0 + �1�, �39�

P�P�� =1

2��0 + �1 cos 2� + �2 sin 2��. �40�

ence the Pauli expansion of the composed polarizer [Eq.37)] is

P =1

4��0 + �1 cos 2� + �2 sin 2� + �1 + �0 cos 2�

− i�3 sin 2��, =1

2cos ��0 + �1�cos � + ��2 − i�3�sin �.

�41�

The Stokes coefficient of �3 is −� /2 out-of-phasequadrature) with those of �1 and �2. The Pauli axis vec-or of the operator (41) is, apart from a constant,

r�cos �, sin �, − i sin ��. �42�

t is an irreducible complex vector; the operator P is aonnormal one.Another sandwich that acts as a non-Hermitian linear

olarizer is constituted by a linear polarizer followed by aalf-wave linear retarder at the azimuth � /2�0,�. Suchn arrangement is used in the half-shade analyzer in po-arimetry. Obviously, the half-wave plate shifts the azi-

uth of the linearly polarized incident light by �. The op-rator of this sandwich is

R�P�/2��P�Px�. �43�

Here the Poincaré axes of the linear polarizer and ofhe half-wave plate are

nP = �1,0,0�,

nR = �cos �,sin �,0�, �44�

nd the Pauli expansion of their operators may be written

P�Px� =1

2��0 + �1�, �45�

R�P�/2�� = i��1 cos � + �2 sin ��. �46�

he composite operator is

i

2��1 cos � + �2 sin ��0 + �1�,

=i

2��0 + �1�cos � + ��2 − i�3�sin �, �47�

hich is very similar to Eq. (41) that is physically justifi-ble: both operators (37) and (43) give �-linearly polarizedight for any input.

Again the Pauli axis vector of the operator is a complexector:

r�cos �,sin �, − i sin ��; �48�

he operator is a nonnormal one.Let us consider now another non-Hermitian polarizer,

he circular polarizer obtained by laminating together ainear polarizer and a linear � /2 retarder with the trans-

ission direction of the polarizer at 45° to the proper axesf the retarder,

C = R�Px��/2�P�P45°�. �49�

The Poincaré axes of the linear retarder and of theuarter-wave plate are, respectively,

nP�0,1,0�,

nR�1,0,0�. �50�

heir operators may be written as

P�P45°�=

1

2��0 + �2�, �51�

R�Px��/2� =1

�2��0 + i�1�, �52�

o that the Pauli expansion of the inhomogeneous circularolarizer operator [Eq. (48)] is

C =1

2�2��0 + i�1 + �2 − �3�. �53�

he Stokes coefficients of �1, �2, and �3 are not all inhase nor in opposition. The Pauli axis vector of the op-rator is complex:

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. Orthogonal Composite Devicese shall deduce now, for comparison, the Pauli algebraic

xpansions of two widespread orthogonal composite de-ices: the transcendent retarder and the Pancharatnam’sHQ variable retarder.It is well known that a succession of two half-wave

lates at a relative azimuth of 45° of their axes acts as a �ircular retarder. This is the simplest transcendentetarder.37 A generalization of this device was given by Ri-hartz and Hsü38 by putting the two half-wave plates at aelative azimuth �,

RT = R�P��R�Px��. �55�

he Poincaré axes of the two half-wave plates are

n�1,0,0�,

n�cos 2�,sin 2�,0�, �56�

o that their operators take on the form

R�Px�� = i�1, �57�

R�P�� = i��1 cos 2� + �2 sin 2��. �58�

hus, the Pauli expansion of the transcendent retarderEq. (53)] is

RT = − ��12 cos 2� + �2�1 sin 2�� = − �I cos 2� − i�3 sin 2��.

�59�

his is a unitary operator with the Poincaré axis

n�0,0, + 1�, �60�

nd that gives a −4� rotation of the incident state of po-arization on the Poincaré sphere about the �R� axis.ence it corresponds to an (orthogonal) circular retarder:

RT = ei�R�R��− 4��. �61�

By analyzing a more general combination of three bire-ringent plates, Pancharatnam39 noted that a combina-ion of two quarter-wave plates with parallel principalxes, between which is placed a half-wave plate with aariable azimuth of the principal axis, gives rise to a vari-ble linear retarder:

RP = R�Px��/2�R�P��R�px��/2�. �62�

Let us express first the R�Px�� /2� quarter-wave platesperators in a Pauli algebraic form. The Poincaré axis ofhe operators is

n�1,0,0�, �63�

nd the angle of the rotation induced by these operatorsn the Poincaré sphere is �=� /2. Hence with Eq. (2) theperators may be written,

R�Px�� =1

�2�I + i�1�. �64�

imilarly, for the half-wave plate of the �-fast axis,�P��, we have

n�cos 2�,sin 2�,0�, �65�

nd �=�. The Pauli algebraic expression of its operator is

R�P�� = i��1 cos 2� + �2 sin 2��. �66�

ith Eqs. (64) and (66) in Eq. (62) we get

RP =i

2��1 cos 2� + �2 sin 2� + iI cos 2� + �3 sin 2�

+ iI cos 2� − �3 sin 2� − �1cos 2� + �2 sin 2��,

RP = − �I cos 2� − i�2 sin 2��. �67�

Apart from a phase factor ei�=−1, Eq. (67) has the formf a unitary operator, with the Poincaré axis

n�0,− 1,0�, �68�

nd inducing a rotation �=4� on the Poincaré sphere.This is a 4� linear retarder of fast-axis −45° (or a −4�

inear retarder of fast-axis 45°):

R�Px��/2�R�P��R�Px��/2� = R�P45°��− 4�� R�P−45°�

�4��.

�69�

From the main viewpoint of our analysis we have tooint out that the Pauli axes of these last two normalunitary) operators, Eqs. (60) and (68), reduce to theireal, Poincaré axes on the Poincaré sphere.

. CONCLUSIONShe Pauli expansions of the orthogonal projector [Eq. (1)]nd of the unitary operator [Eq. (2)] are currently handleds device operators of the canonical ideal polarizers andetarders, respectively, in polarization optics, where thesexpansions have been established inductively, on an ex-erimental ground.Generally, the polarization devices are not of the or-

hogonal kind. The eigenvectors of propagating light inrystals are, generally (and naturally), nonorthogonal.onnormal operators govern the light propagation in

rystals, particularly in polarization devices, in the mosteneral case. This is one of the reasons why the interestn nonnormal operators in physics is growing.14–23

Starting from this very definite interest, we have givenunified theory of the Pauli algebraic development of

arious normal and nonnormal operators. In this frame-ork, the Pauli expressions of the orthogonal projectorsnd unitary operators are obtained in a hierarchic deduc-ive way, together with other kinds of normal and nonnor-al operators.In this approach, to each operator defined on a complex

ector space of two dimensions over the field of complexumbers C1 corresponds a vector in C3 that we call theauli axis of the operator. In particular, for normal opera-

ors their Pauli axes are real vectors or are reducible by a

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hase shift to real vectors, whereas the Pauli axes of theonnormal operators are irreducible complex vectors.Bearing in mind the great interest manifested in recent

ears, in a large variety of physical problems, for nonnor-al operators (defined as “non-Hermitian” or “nonuni-

ary” and handled especially in matrix forms), we hopehat our results will be useful in the mathematical ap-roach to a larger field of physical problems.Well known, especially in the past decade, the group

heory of SL�2,c� and of some of its subgroups was exten-ively applied in various fields of classical and quantumptics, e.g., ray optics,40 beam propagation through first-rder systems,41 analysis of the states of light with orbitalngular momentum,42 polarization optics,43 multilayerptics,44,45 interferometry,46 and coherent and squeezedtates of light.47 Bearing in mind that SL�2,c� is locallysomorphic to the six-parameter Lorentz group SO�3,1�, ahysical system that can be analyzed in terms of SL�2,c�anguage can be equally explained in the language of theorentz group.48

From the viewpoint of the group theory, the non-ermitian polarizer [Eq. (37)] is an example of Wigner ro-

ation: The two Hermitian polarizers of a sandwich suchs Eq. (37) are described by squeeze operators, and theirroduct is not a squeeze operator, but a squeeze operatorollowed or preceded by a rotation.

We think that, generally, the group-theoretical ap-roach has not, until now, paid special attention to the di-hotomy of normal and nonnormal operators, perhaps be-ause the normal operators—although a very importantlass—do not form a group. This problem has appearedrom another field of interest in physics,14–31and probablyhe group-theoretical approach will have an importantord to say in this concern.

CKNOWLEDGMENTSe thank the reviewers for comments that resulted in the

ubstantial improvement of the manuscript.

The authors’ email addresses are [email protected] [email protected].

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pauli algebraic forms of normal and nonnormal operators

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