DOCSLIB.ORG

The Unitary Irreducible Representations of SL(2, R) in All Subgroup Reductions A)

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Unitary Irreducible Representations of SL(2, R) in All Subgroup Reductions A) The unitary irreducible representations of SL(2, R) in all subgroup reductions a) Debabrata Basu b) and Kurt Bernardo Wolf lnstituto de Investigaciones en Matematicas Aplicadas yen Sistemas (IIMAS). Universidad Nacional Aut6noma de Mexico. Apdo. Postal 20-726 Mexico 20. D. F.. Mexico (Received 23 December 1980; accepted for publication 5 1 une 1981) We use the canonical transform realization of SL(2, R ) in order to find all matrix elements and integral kernels for the unitary irreducible representations of this group. Explicit results are given for all mixed bases and subgroup reductions. These provide the full multiparameter set of integral transforms and series expansions associated to SL(2, R ). P ACS numbers: 02.20.Df 1. INTRODUCTION The complete classification of the Unitary Irreducible called canonical transforms. IX.2X It is unique in that the asso­ representations (UIRs) of the three-dimensional Lorentz ciated Lie algebra is an algebra of second-order differential group SO(2, 1) and of its twofold covering group SL(2, R ) operators on a dense common domain in these Hilbert were given by Bargmann in his classic 1947 article,1 where spaces. The action is thus distinct from-although unitarily one can find the UIR matrix elements-rows and columns equivalent20.21 to-the SL(2, R) action as a Lie transforma­ classified by the UIRs of the compact subgroup SO(2)-in tion group on coset spaces, of the Lie-Bargmann multiplier L2 representations29 on the unit circle or disk. explicit form. This group, its covering groups SO(2, 1)~ The canonical transform realization has provided a de­ 1 00 ---- l gree of uniformity in the treatment of the discrete series '! of SUr 1, 1)~Sp(2, R );:::; SL(2, R ) ~ SL(2, R ) and its represen- UIRs on the one hand and the continuous series21 of UIRs 2 tations were further studied by Barut and Fronsdal, Pu­ on the other. In this article it has enabled us to evaluate, in a 5 kanski/ Sally, 1r.,4 and in a book by Lang. straightforward and unified way, the UIR matrix elements The study of group representations in different bases is and integral kernels of finite SL(2, R ) elements. In contrast of interest both from the mathematical and the physical with some of the previous investigations, this approach deals point of view. The intimate connections between the repre­ with the general SL(2, R ) group element, rather than with sentations of Lie groups and the special functions of math­ specific one-parameter subgroups. Although Bargmann's ematical physics have long been recognized and treated in results on UIRs ofSL(2, R ) in the compact subgroup basis30 6 textbooks. In physics, subgroup reductions corresponding are well known, it is also true that other continuous noncom­ to different bases of the Lorentz and other groups lead to pact and mixed-basis reductions have so far not received various ways to correlate or interpret data, as in the descrip­ uniform consideration2.9.lo.12,31-13 and are scattered in the tion of the high-energy scattering dynamics,7 which requires literature. The discrete series ofUIRs in all subgroup reduc­ the reduction SO(2, 1):J SOt 1, 1) among others. This interest tions was undertaken by Boyer and W 01(,4 using canonical coincided with the investigations ofMukunda,R-11 Barut,2.12 transforms. We repeat their results here since the journal is Lindblad and Nagel, 13 and others, who analyzed this chain not generally available and the article contains some errata. in some detail and computed the generalized representation The mixed-basis matrix elements of the continuous series matrices (or integral kernels) of one-parameter subgroups were treated by Kalnins,31 who gave expressions for one­ and found the coupling coefficients. parameter subgroups in terms of Whittaker and Laguerre In the study of the role of canonical transformations in functions of the second kind. 35 All our expressions are given quantum mechanics, the work of Moshinsky and l4 15 in terms of confluent and Gauss hypergeometric functions, Quesne . started from linear transformations between co­ and have uniformity of notation, normalization, and phase ordinate and momentum observables and lead to the oscilla­ conventions. The purpose of this paper is to give a compre­ tor (metaplectic) representation ofSp(2, R ). In contrast to the hensive derivation and listing of all subgroup reductions. realizations given by Bargmann 1 and by Gel'fand et ai., 16 in The plan of the article is as follows. In Sec. 2 we display which the group acts as a Lie transformation group on func­ the needed formulas from the theory of canonical transforms tions of a coset manifold, the group actions in the construc­ for the general method of construction and, since we want to tions of Moshinsky, 14.15.17 Seligman, Wolf, IR-23 Burdet, Per­ describe all VIR matrix elements and integral kernels, we rin and Perroud,24 and present in the work of others, 25-27 is organize the notation properly in due accordance with Barg­ an integral transform realization ofSL(2, R ) on y2(R ) Hil­ mann's conventions. In Sec. 3 and 4 we give the results for bert spaces. This group of integral transforms has been the discrete and continuous (nonexceptional and exception­ al) representation series. The first subsection of each lists the subgroup-adapted basis functions, the second treats the a)Work performed under financial assistance from Consejo Nacional de Ciencia y Tecnologia (CONACYTI Project ICCBIND 790370. mixed-basis expressions, while the third subsection treats "'On leave from Department of Physics, Indian Institute of Technology. the subgroup reductions, i.e., the cases when the row and Kharagpur 721302. India. column variables refer to the same subgroup. These are ex- 189 J. Math. Phys. 23(2). February 1982 0022·2488/82/020189-17$02.50 © 1982 American Institute of Physics 189 Downloaded 28 Jun 2011 to 132.248.33.126. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions pressed as Gauss or confluent hypergeometric functions 3o 37 and, alternatively, as cylinder and Whittaker functions • of the three independent SL(2, R ) parameters. Certain cases where the symbol SA 'EA stands for summation in the case of of interest are pointed out in a further subsection. Compari­ proper, and integration in the case of generalized, bases, The son with alternative derivations available in the literature is unitarity and irreducibility properties ofD follow from simi­ pointed out whenever we are aware of such results. lar requirements for the action (2.1) on JY. The representation matrix elements for the compact The reasons for which this straightforward program of­ subgroup chain were obtained by Bargmann as solutions to ten fails to provide a definite result have to do more with 38 differential equations with boundary conditions imposed knowing the "best" choice of basis functions! ifJ). (r) l AEA and by the group identity. We come to the evaluation of an inte­ the problem of explicit computation of the integral in (2.2b), gral as the last step to the same end. We make use of a meth­ than with matters of principle. The bases are usually chosen od by Majumdar and Basu32 on hypergeometric series Mel­ as the eigenvectors of one or more operators in the Lie alge­ lin expansions to solve three of the six chains in each series. bra-so that subgroup reductions result-while the space In the special case of the continuous series in the compact JY' is an !f 2(X) space on a coset manifold X = G / H (or subgroup reduction, such an integral (a Gaussian of imagi­ H \. G) with some convenient subgroup He G. A closely re­ nary width times two Whittaker functions, one with a res­ lated approach to part (ii) of evaluation of (2.2b) calls for caled argument) is not available in the literature. Through (ii') finding these functions for various one-parameter Bargmann's result this is evaluated. subgroups of G as solutions of differential equations ob­ In Sec. 5 we point out that the six different mixed-basis tained from the subgroup generators, subject to the bound­ and subgroup-reduced representation matrix elements con­ ary conditions D(e) = 1 at the group identity eEG. stitute six families of SL(2, R ) integral and discrete trans­ The group G which we consider here is SL(2, R ): forms, as well as series expansions, of which the set of ca­ nonical transforms is but one. The Appendix summarizes {g = e!)la,b,c,dER, det g= I}. (2.3) some information about the groups SU(1, 1), SL(2, R ), and Starting with Bargmann I a number of authors have imple­ their UIRs as classified by Bargmann. Throughout this arti­ mented the program (i)-(ii) or (i)-(ii'), using for the support­ cle Z and R stand for the set of integers and real numbers. ing space X the coset space provided by the I wasawa decom­ Boldfaced symbols indicate vectors or matrices. For brevity, position NA \.NAK = SI (i,e., the circle) and Bargmann's we shall speak ofUIR matrix elements encompassing both multiplier action. 29 This is unitary in !f2(SI) for the continu­ the ordinary and generalized (i.e., integral transform kernel) 29 ous non exceptional representation series ; for the continu­ cases. 2 ous exceptional and discrete series it is 'y n c(SI) and As a general observation, we should remark that the 39 0 y2 n o(Sd with non local measures .4 n C and n D, The lat­ canonical transform realization ofSL(2, R ) can be regarded ter is equivalent20 to a space of analytic functions on the unit as a complementary alternative to Bargmann's treatment of 29 the same group.
Load more

Recommended publications
  • The Unitary Representations of the Poincaré Group in Any Spacetime
    The unitary representations of the Poincar´e group in any spacetime dimension Xavier Bekaert a and Nicolas Boulanger b a Institut Denis Poisson, Unit´emixte de Recherche 7013, Universit´ede Tours, Universit´ed’Orl´eans, CNRS, Parc de Grandmont, 37200 Tours (France) [email protected] b Service de Physique de l’Univers, Champs et Gravitation Universit´ede Mons – UMONS, Place du Parc 20, 7000 Mons (Belgium) [email protected] An extensive group-theoretical treatment of linear relativistic field equa- tions on Minkowski spacetime of arbitrary dimension D > 2 is presented in these lecture notes. To start with, the one-to-one correspondence be- tween linear relativistic field equations and unitary representations of the isometry group is reviewed. In turn, the method of induced representa- tions reduces the problem of classifying the representations of the Poincar´e group ISO(D 1, 1) to the classification of the representations of the sta- − bility subgroups only. Therefore, an exhaustive treatment of the two most important classes of unitary irreducible representations, corresponding to massive and massless particles (the latter class decomposing in turn into the “helicity” and the “infinite-spin” representations) may be performed via the well-known representation theory of the orthogonal groups O(n) (with D 4 <n<D ). Finally, covariant field equations are given for each − unitary irreducible representation of the Poincar´egroup with non-negative arXiv:hep-th/0611263v2 13 Jun 2021 mass-squared. Tachyonic representations are also examined. All these steps are covered in many details and with examples. The present notes also include a self-contained review of the representation theory of the general linear and (in)homogeneous orthogonal groups in terms of Young diagrams.
    [Show full text]
  • 1 the Spin Homomorphism SL2(C) → SO1,3(R) a Summary from Multiple Sources Written by Y
    1 The spin homomorphism SL2(C) ! SO1;3(R) A summary from multiple sources written by Y. Feng and Katherine E. Stange Abstract We will discuss the spin homomorphism SL2(C) ! SO1;3(R) in three manners. Firstly we interpret SL2(C) as acting on the Minkowski 1;3 spacetime R ; secondly by viewing the quadratic form as a twisted 1 1 P × P ; and finally using Clifford groups. 1.1 Introduction The spin homomorphism SL2(C) ! SO1;3(R) is a homomorphism of classical matrix Lie groups. The lefthand group con- sists of 2 × 2 complex matrices with determinant 1. The righthand group consists of 4 × 4 real matrices with determinant 1 which preserve some fixed real quadratic form Q of signature (1; 3). This map is alternately called the spinor map and variations. The image of this map is the identity component + of SO1;3(R), denoted SO1;3(R). The kernel is {±Ig. Therefore, we obtain an isomorphism + PSL2(C) = SL2(C)= ± I ' SO1;3(R): This is one of a family of isomorphisms of Lie groups called exceptional iso- morphisms. In Section 1.3, we give the spin homomorphism explicitly, al- though these formulae are unenlightening by themselves. In Section 1.4 we describe O1;3(R) in greater detail as the group of Lorentz transformations. This document describes this homomorphism from three distinct per- spectives. The first is very concrete, and constructs, using the language of Minkowski space, Lorentz transformations and Hermitian matrices, an ex- 4 plicit action of SL2(C) on R preserving Q (Section 1.5).
    [Show full text]
  • Physics of the Lorentz Group
    Physics of the Lorentz Group Sibel Ba¸skal Department of Physics, Middle East Technical University, 06800 Ankara, Turkey e-mail: [email protected] Young S. Kim Center for Fundamental Physics, University of Maryland, College Park, Maryland 20742, U.S.A. e-mail: [email protected] Marilyn E. Noz Department of Radiology, New York University, New York, NY 10016 U.S.A. e-mail: [email protected] To be published in the IOP Concise Book Series. Preface When Newton formulated his law of gravity, he wrote down his formula applicable to two point particles. It took him 20 years to prove that his formula works also for extended objects such as the sun and earth. When Einstein formulated his special relativity in 1905, he worked out the transformation law for point particles. The question is what happens when those particles have space-time extensions. The hydrogen atom is a case in point. The hydrogen atom is small enough to be regarded as a particle obeyingp Einstein's law of Lorentz transformations including the energy-momentum relation E = p2 + m2. Yet, it is known to have a rich internal space-time structure, rich enough to provide the foundation of quantum mechanics. Indeed, Niels Bohr was interested in why the energy levels of the hydrogen atom are discrete. His interest led to the replacement of the orbit by a standing wave. Before and after 1927, Einstein and Bohr met occasionally to discuss physics. It is possible that they discussed how the hydrogen atom with an electron orbit or as a standing-wave looks to moving observers.
    [Show full text]
  • Physics of the Lorentz Group
    Physics of the Lorentz Group Physics of the Lorentz Group Sibel Ba¸skal Department of Physics, Middle East Technical University, 06800 Ankara, Turkey e-mail: [email protected] Young S. Kim Center for Fundamental Physics, University of Maryland, College Park, Maryland 20742, U.S.A. e-mail: [email protected] Marilyn E. Noz Department of Radiology, New York University, New York, NY 10016 U.S.A. e-mail: [email protected] Preface When Newton formulated his law of gravity, he wrote down his formula applicable to two point particles. It took him 20 years to prove that his formula works also for extended objects such as the sun and earth. When Einstein formulated his special relativity in 1905, he worked out the transfor- mation law for point particles. The question is what happens when those particles have space-time extensions. The hydrogen atom is a case in point. The hydrogen atom is small enough to be regarded as a particle obeying Einstein'sp law of Lorentz transformations in- cluding the energy-momentum relation E = p2 + m2. Yet, it is known to have a rich internal space-time structure, rich enough to provide the foundation of quantum mechanics. Indeed, Niels Bohr was interested in why the energy levels of the hydrogen atom are discrete. His interest led to the replacement of the orbit by a standing wave. Before and after 1927, Einstein and Bohr met occasionally to discuss physics. It is possible that they discussed how the hydrogen atom with an electron orbit or as a standing-wave looks to moving observers.
    [Show full text]
  • Quantum Mechanics 2 Tutorial 9: the Lorentz Group
    Quantum Mechanics 2 Tutorial 9: The Lorentz Group Yehonatan Viernik December 27, 2020 1 Remarks on Lie Theory and Representations 1.1 Casimir elements and the Cartan subalgebra Let us first give loose definitions, and then discuss their implications. A Cartan subalgebra for semi-simple1 Lie algebras is an abelian subalgebra with the nice property that op- erators in the adjoint representation of the Cartan can be diagonalized simultaneously. Often it is claimed to be the largest commuting subalgebra, but this is actually slightly wrong. A correct version of this statement is that the Cartan is the maximal subalgebra that is both commuting and consisting of simultaneously diagonalizable operators in the adjoint. Next, a Casimir element is something that is constructed from the algebra, but is not actually part of the algebra. Its main property is that it commutes with all the generators of the algebra. The most commonly used one is the quadratic Casimir, which is typically just the sum of squares of the generators of the algebra 2 2 2 2 (e.g. L = Lx + Ly + Lz is a quadratic Casimir of so(3)). We now need to say what we mean by \square" 2 and \commutes", because the Lie algebra itself doesn't have these notions. For example, Lx is meaningless in terms of elements of so(3). As a matrix, once a representation is specified, it of course gains meaning, because matrices are endowed with multiplication. But the Lie algebra only has the Lie bracket and linear combinations to work with. So instead, the Casimir is living inside a \larger" algebra called the universal enveloping algebra, where the Lie bracket is realized as an explicit commutator.
    [Show full text]
  • Symmetry Groups in Physics Part 2
    Symmetry groups in physics Part 2 George Jorjadze Free University of Tbilisi Zielona Gora - 25.01.2017 G.J.| Symmetry groups in physics Lecture 3 1/24 Contents • Lorentz group • Poincare group • Symplectic group • Lie group • Lie algebra • Lie algebra representation • sl(2; R) algebra • UIRs of su(2) algebra G.J.| Symmetry groups in physics Contents 2/24 Lorentz group Now we consider spacetime symmetry groups in relativistic mechanics. The spacetime geometry here is given by Minkowski space, which is a 4d space with coordinates xµ, µ = (0; 1; 2; 3), and the metric tensor gµν = diag(−1; 1; 1; 1). We set c = 1 (see Lecture 1) and x0 is associated with time. Lorentz group describes transformations of spacetime coordinates between inertial systems. Mathematically it is given by linear transformations µ µ µ ν x 7! x~ = Λ n x ; which preserves the metric structure of Minkowski space α β Λ µ gαβ Λ ν = gµν : The matrix form of these equations reads ΛT g L = g : This relation provides 10 independent equations, since g is symmetric. Hence, Lorentz group is 6 parametric. G.J.| Symmetry groups in physics Lorentz group 3/24 From the definition of Lorentz group follows that 0 2 i i det[Λ] = ±1 and (Λ 0) − Λ 0 Λ 0 = 1. Therefore, Lorentz group splits in four non-connected parts 0 0 1: (det[Λ] = 1; Λ 0 ≥ 1); 2: (det[Λ] = 1; Λ 0 ≤ 1), 0 0 3: (det[Λ] = −1; Λ 0 ≥ 1); 4: (det[Λ] = −1; Λ 0 ≤ 1). The matrices of the first part are called proper Lorentz transformations.
    [Show full text]
  • Group Theory - QMII 2017
    Group Theory - QMII 2017 There are many references about the subject. Here are three of them: 1. The Quantum Theory of Fields Vol. I,S.Weinberg. 2. Quantum Field Theory,L.Ryder. 3. Group Theory in Physics, W. K. Tung. 1 The proper Lorentz group and poincare 1.1 Reminder Recall that at the end of the day spacial relativity is a theory of transfor- mation rules. The allowed set of transformations are those which leaves the interval invariant. The interval which is measured to be the same in all inertial frames of reference is 2 µ ⌫ ds = ⌘µ⌫dx dx . (1) We found that under the general transformation dxµ ⇤µ dx⌫,theallowed ! ⌫ ⇤’s are those who satisfies the following condition: ⌘ =⇤T ⌘⇤ . (2) We separate them to two classes - rotations and boosts. Note that in both cases the determinant is det(⇤) = 1. This is related to the fact that this continuous transformations are connected to the identity, i.e. ⇤(0, 0) = 1. 1.2 Discrete symmetries The group O(1, 3) contains also transformations with ⇤ = 1. This is | | − related to discrete symmetries. On top of the continuous rotations and boosts there are two additional discrete transformations - time reversal (T )and 1 parity (P ) 1000 10 0 0 − 0 01001 00 10 01 T = ,P= − . (3) B 0010C B00 10C B C B − C B 0001C B00 0 1C B C B − C @ A @ A All together we get the complete Lorentz group which has four discon- nected pieces P L+" L" = P L+" proper-orthochronous ! improper− orthochronous· − T T l L+# = T L+" L#l= P T L+" proper nonorthochronous· P! improper− nonorthochronous· · − − From now on we will consider only the proper-orthochronous Lorentz group L+" .
    [Show full text]
  • Exceptional Structures in Mathematics and Physics and the Role of the Octonions∗
    CBPF-NF-042/03 Exceptional Structures in Mathematics and Physics and the Role of the Octonions∗ Francesco Toppan Centro Brasileiro de Pesquisas F´ısicas - CCP Rua Dr. Xavier Sigaud 150, 22290-180, Rio de Janeiro (RJ), Brazil Abstract There is a growing interest in the logical possibility that exceptional mathematical structures (exceptional Lie and superLie algebras, the exceptional Jordan algebra, etc.) could be linked to an ultimate “exceptional” formulation for a Theory Of Everything (TOE). The maximal division algebra of the octonions can be held as the mathematical responsible for the existence of the exceptional structures mentioned above. In this context it is quite motivating to systematically investigate the properties of octonionic spinors and the octonionic realizations of supersymmetry. In particular the M-algebra can be consistently defined for two structures only, a real structure, leading to the standard M- algebra, and an octonionic structure. The octonionic version of the M-algebra admits striking properties induced by octonionic p-forms identities. Key-words: Exceptional Lie algebras; Octonions; M-theory. ∗Proceedings of the International Workshop “Supersymmetries and Quantum Symmetries” (SQS’03, 24-29 July, 2003), Dubna, Russia. CBPF-NF-042/03 1 1 Introduction. The search of an ultimate “Theory Of Everything” (TOE) corresponds to a very ambitious and challenging program. At present the conjectured M-Theory (whose dynamics is yet to unravel), a non-perturbative theory underlying the web of dualities among the five consistent superstring theories and the maximal eleven-dimensional supergravity, is regarded by perhaps the majority of physicists as the most promising candidate for a TOE [1].
    [Show full text]
  • Representations of Lorentz and Poincaré Groups
    Representations of Lorentz and Poincar´egroups Joseph Maciejko CMITP and Department of Physics, Stanford University, Stanford, California, 94305 I. LORENTZ GROUP We consider first the Lorentz group O(1, 3) with infinitesimal generators J µν and the associated Lie algebra given by [J µν,J ρσ] = i(gνρJ µσ − gµρJ νσ − gνσJ µρ + gµσJ νρ) (1) In quantum field theory, we actually consider a subgroup of O(1, 3), the proper orthochronous + 0 or restricted Lorentz group SO (1, 3) = {Λ ∈ O(1, 3)| det Λ = 1 and Λ 0 ≥ 0}, which ex- cludes parity and time-reversal transformations that are thus considered as separate, discrete operations P and T . A generic element Λ of the Lorentz group is given by exponentiating 1 µν the generators together with the parameters of the transformation, Λ = exp(−iωµνJ /2). The Lorentz group has both finite-dimensional and infinite-dimensional representations. However, it is non-compact, therefore its finite-dimensional representations are not unitary (the generators are not Hermitian). The generators of the infinite-dimensional representa- tions can be chosen to be Hermitian. A. Finite-dimensional representations We first study the finite-dimensional representations of SO+(1, 3). These representations act on finite-dimensional vector spaces (the base space). Elements of these vector spaces are said to transform according to the given representation. Trivial representation. In the trivial representation, we have the one-dimensional representation J µν = 0. Hence any Lorentz transformation Λ is represented by 1. This representation acts on a one-dimensional vector space whose elements are 1-component ob- jects called Lorentz scalars.
    [Show full text]
  • Chapter 1 Lorentz Group and Lorentz Invariance
    Chapter 1 Lorentz Group and Lorentz Invariance In studying Lorentz-invariant wave equations, it is essential that we put our under- standing of the Lorentz group on firm ground. We first define the Lorentz transfor- mation as any transformation that keeps the 4-vector inner product invariant, and proceed to classify such transformations according to the determinant of the transfor- mation matrix and the sign of the time component. We then introduce the generators of the Lorentz group by which any Lorentz transformation continuously connected to the identity can be written in an exponential form. The generators of the Lorentz group will later play a critical role in finding the transformation property of the Dirac spinors. 1.1 Lorentz Boost Throughout this book, we will use a unit system in which the speed of light c is unity. This may be accomplished for example by taking the unit of time to be one second and that of length to be 2.99792458 × 1010 cm (this number is exact1), or taking the unit of length to be 1 cm and that of time to be (2.99792458 × 1010)−1 second. How it is accomplished is irrelevant at this point. Suppose an inertial frame K (space-time coordinates labeled by t, x, y, z)ismov- ing with velocity β in another inertial frame K (space-time coordinates labeled by t,x,y,z)asshown in Figure 1.1. The 3-component velocity of the origin of K 1One cm is defined (1983) such that the speed of light in vacuum is 2.99792458 × 1010 cm per second, where one second is defined (1967) to be 9192631770 times the oscillation period of the hyper-fine splitting of the Cs133 ground state.
    [Show full text]
  • Poincaré Group
    Appendix B Poincar´egroup Poincar´einvariance is the fundamental symmetry in particle physics. A relativistic quantum field theory must have a Poincar´e-invariant action. This means that its fields must transform under representations of the Poincar´egroup and Poincar´einvariance must be implemented unitarily on the state space. Here we will collect some properties of the Lorentz and Poincar´egroups together with their representation theory. B.1 Lorentz and Poincar´egroup Lorentz group. We work in Minkowski space with the metric tensor g = (gµν) = diag (1; 1; 1; 1), where the scalar product is given by − − − x y := xT g y = x0y0 x y = g xµyν = x yµ: (B.1) · − · µν µ Instead of carrying around explicit instances of g, it is more convenient to use the index notation where upper and lower indices are summed over. Lorentz transformations are those transformations x0 = Λx that leave the scalar product invariant: (Λx) (Λy) = x y xT ΛT g Λ y = xT g y ΛT g Λ = g : (B.2) · · ) ) Written in components, this condition takes the form µ ν gαβ = gµν Λ α Λ β : (B.3) Since the metric tensor is symmetric, this gives 10 constraints; the Lorentz transfor- mation Λ is a 4 4 matrix, so it depends on 16 10 = 6 independent parameters. If × α − α α we write an infinitesimal transformation as Λ β = δ β + " β + ::: , then it follows from Eq. (B.3) that " = " must be totally antisymmetric. αβ − βα The transformations of a space with coordinates y : : : y ; x : : : x that leave the f 1 n 1 mg quadratic form (y2 + + y2) (x2 + + x2 ) invariant constitute the orthogonal 1 ··· n − 1 ··· m group O(m; n), so the Lorentz group is O(3; 1).
    [Show full text]
  • REPRESENTATIONS of the POINCARE GROUP for QUANTUM FIELD THEORY by James Kettner
    REPRESENTATIONS OF THE POINCARE GROUP FOR QUANTUM FIELD THEORY by James Kettner The uni…cation of quantum mechanics and special relativity into quantum …eld theory still contains some of the major assumptions of non-relativistic quan- tum mechanics. In particular, it is still postulated that a physical state corre- sponds to a vector in a Hilbert space up to multiplication by a complex number, i.e. states correspond to rays in a Hilbert space. The probability of a transition from a state represented by a ray = ; where is a vector in the Hilbert f g space and = 0 is a complex number, to a state represented by = is given by the6 ray product b f g b (; 2 [ ] = j j j (; )( ; ) where (; ) is the inner productb b on the Hilbert space; this de…nition does not depend on the choice of vectors in and in . Therefore the transfor- mations that preserve probabilities are only de…ned up to a complex factor. Everybody can safely ignore this complicationb because,b in 1931, Eugene Wigner proved that any transformation preserving probabilities can be replaced with either a linear, unitary transformation or an anti-linear, anti-unitary transfor- mation. In 1939, Wigner published a paper with important results relating quantum mechanics and special relativity. These results are: 1. A Hilbert space of states carries a representation, up to a factor, of the inhomogeneous Lorentz group by linear, unitary operators. Wigner states that "... there corresponds to every invariant quantum mechanical system of equations such a representation of the inhomogeneous Lorentz group. This representation, on the other hand, though not su¢ cient to replace quantum mechanical operators entirely, can replace them to a large ex- tent." 2.
    [Show full text]