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G. Cassinelli, E. De Vito, P. J. Lahti, A. Levrero
The Theory
of Symmetry Actions
in Quantum Mechanics
with an Application to the Galilei Group
123
Authors
GianniCassinelli PekkaJ.Lahti
UniversitadiGenova UniversityofTurku
DipartimentodiFisica DepartmentofPhysics
16146Genova,Italy 20014Turku,Finland
ErnestoDeVito AlbertoLevrero
DipartimentodiMatematica UniversitadiGenova
PuraedApplicata"G.Vitali" DipartimentodiFisica
41000Modena,Italy 16146Genova,Italy
G. Cassinelli, E. De Vito, P. J. Lahti, A. Levrero, The Theory of Symmetry Actions in
Quantum Mechanics, Lect. Notes Phys. 654 (Springer, Berlin Heidelberg 2004), DOI
10.1007/b99455
LibraryofCongressControlNumber:2004110193
ISSN0075-8450
ISBN3-540-22802-0SpringerBerlinHeidelbergNewYork
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Group
Preface
This book is devoted to the study of the symmetries in quantum mechanics.
Inmanyelementaryexpositionsofquantumtheory,oneofthebasicassump-
tions is that a group G of transformations is a group of symmetries for a
quantumsystemifGadmitsaunitaryrepresentationU actingontheHilbert
space H associated with the system. The requirement that, given g ∈G, the
corresponding operator U is unitary is motived by the need for preserving
g
the transition probability between any two vector states ϕ,ψ ∈H,
|(cid:2)ϕ,U ψ(cid:3)|2 =|(cid:2)ϕ,ψ(cid:3)|2. (0.1)
g
The composition law
U =U U (0.2)
g1g2 g1 g2
encodes the assumption that the physical symmetries form a group of trans-
formations on the set of vector states.
However, as soon as one considers some explicit application, the above
frameworkappearstoorestrictive.Forexample,itiswellknownthatthewave
function ϕ of an electron changes its sign under a rotation of 2π; the Dirac
equation is not invariant under the Poincar´e group, but under its universal
coveringgroup;theSchro¨dingerequationisinvariantneitherundertheGalilei
group nor under its universal covering group.
The above pathologies have important physical consequences: bosons and
fermions can not be coherently superposed, the canonical position and mo-
mentum observables of a Galilei invariant particle do not commute and par-
ticles with different mass cannot be coherently superposed.
For the Poincar´e group the above problem was first solved by Wigner
in his seminal paper [40] and it was systematically studied by Bargmann,
[1], and Mackey, [27] (see, also, the book of Varadarajan, [35], for a detailed
exposition of the theory).
These authors clarified that in order to preserve (0.1), one only has to
require that U be either unitary or antiunitary and (0.2) can be replaced by
the weaker condition
U =m(g ,g )U U , (0.3)
g1g2 1 2 g1 g2
VIII Preface
where m(g ,g ) is a complex number of modulo one (U is said to be a pro-
1 2
jective representation). Moreover, they showed that the study of projective
representations can be reduced to the theory of ordinary unitary representa-
tionsbyenlargingthephysicalgroupofsymmetries.Forexample,therotation
group SO(3) has to be replaced by its universal covering group SU(2). The
trick of replacing the physical symmetry group G with its universal covering
groupG∗ issowellknowninthephysicscommunitythatthegroupG∗ itself
is considered as the true physical symmetry group. However, for the Galilei
groupthecoveringgroupisnotenoughandoneneedsevenalargergroupG,
namely the universal central extension, in order that the unitary (ordinary)
representations of G exhaust all the possible projective representations of G.
Theaimofthisbookistopresentthetheoryneededtoconstructtheuni-
versalcentralextensionfromthephysicalsymmetrygroupinaunified,simple
and mathematically coherent way. Most of the results presented are known.
However, we hope that our exposition will help the reader to understand the
role of the mathematical objects that are introduced in order to take care of
the true projective character of the representations in quantum mechanics.
Finally,ourconstructionofGisveryexplicitandcanbeperformedbysimple
linear algebraic tools. This theory is presented in Chap. 3.
Coming back to (0.1), this equality means that we regard symmetries as
mathematical objects that preserve the transition probability between pure
states.Thestructureoftransitionprobabilityisonlyoneofthevariousphys-
ically relevant structures associated with a quantum system. Other relevant
structures being, for instance, the convex structures of the sets of states and
effects, the order structure of effects, and the algebraic structure of observ-
ables. Therefore it is natural to define symmetry as a bijective map that
preserves one of these structures. In Chap. 2 we present several possibilities
of modeling a symmetry and we show that they all coincide. Hence one may
speakofsymmetriesofaquantumsystem.Thesetofallpossiblesymmetries
forms a topological group Σ and, given a group G, a symmetry action is
defined as a continuous map σ from G to Σ such that
σ =σ σ .
g1g2 g1 g1
As an application of these ideas, in Chaps. 4 and 5 we treat in full detail
the case of the Galilei group both in 3+1 and in 2+1 dimensions. The
choice of the Galilei group instead of the Poincar´e group is motivated first of
all by the fact that the Poincar´e group has already been extensively studied
in the literature. Secondly, from a mathematical point of view, the Galilei
group shows all the pathologies cited above and one needs the full theory
of projective representations. We also treat the 2+1 dimensional case since
thereisanincreasinginterestinthesurfacephenomenabothfromtheoretical
and from experimental points of view.
The last chapter of the book is devoted to the study of Galilei invariant
wave equations. Within the framework of the first quantisation, the need for
wave equations naturally arises if one introduces the interaction of a particle
Preface IX
with a (classical) electromagnetic field by means of the minimal coupling
principle. To this aim, one has to describe the vector states as functions on
the space-time satisfying a differential equation, the wave equation, which is
invariant with respect to the universal central extension of the Galilei group.
In Chap. 6 we describe how these wave equations can be obtained without
using Lagrangian (classical) techniques. In particular, we prove that for a
particle of spin j there exists a linear wave equation, like the Dirac equation
forthePoincar´egroup,suchthattheparticleacquiresagyromagneticinternal
moment with the gyromagnetic ratio 1.
j
Since the book is devoted to the application of the abstract theory to the
Galilei group, we always assume that the symmetry group G is a connected
Lie group. In particular, we do not consider the problem of discrete symme-
tries. In the Appendix we recall some basic mathematical definitions, facts,
and theorems needed in this book. The reader will find them as entries in
the Dictionary of Mathematical Notions in the Appendix. The statement of
definitions and results are usually not given in their full generality but they
are adjusted to our needs.
Contents
1 A Synopsis of Quantum Mechanics ....................... 1
1.1 The Set S of States and the Set P of Pure States ........... 2
1.2 The Set E of Effects and the Set D of Projections........... 3
1.3 Observables ............................................ 5
2 The Automorphism Group of Quantum Mechanics ....... 7
2.1 Automorphism Groups of Quantum Mechanics ............. 7
2.1.1 State Automorphisms ............................. 7
2.1.2 Vector State Automorphisms ....................... 9
2.1.3 Effect Automorphisms............................. 11
2.1.4 Automorphisms on D ............................. 14
2.1.5 Automorphisms of H .............................. 18
2.2 The Wigner Theorem.................................... 19
2.2.1 The Theorem..................................... 19
2.3 The Group Isomorphisms ................................ 23
2.3.1 Isomorphisms .................................... 23
2.3.2 Homeomorphisms................................. 24
2.3.3 The Automorphism Group of Quantum Mechanics .... 25
3 The Symmetry Actions and Their Representations ....... 27
3.1 Symmetry Actions of a Lie Group......................... 28
3.2 Multipliers for Lie Groups................................ 31
3.3 Universal Central Extension of a Connected Lie Group ...... 33
3.4 The Physical Equivalence for Semidirect Products........... 42
3.5 An Example: The Temporal Evolution of a Closed System ... 46
4 The Galilei Groups ....................................... 49
4.1 The 3+1 Dimensional Case.............................. 49
4.1.1 Physical Interpretation ............................ 50
4.1.2 The Covering Group .............................. 50
4.1.3 The Lie Algebra .................................. 51
4.1.4 The Multipliers for the Covering Group.............. 52
4.1.5 The Universal Central Extension.................... 53
4.2 The 2+1 Dimensional Case.............................. 56
XII Contents
4.2.1 The Multipliers for the Covering Group
and the Universal Central Extension ................ 56
5 Galilei Invariant Elementary Particles .................... 61
5.1 The Relativity Principle for Isolated Systems ............... 61
5.1.1 Galilei Systems in Interaction ...................... 63
5.2 Symmetry Actions in 3+1 Dimensions .................... 64
5.2.1 The Dual Group and the Dual Action ............... 64
5.2.2 The Orbits and the Orbit Classes ................... 65
5.2.3 Representations Arising from O(cid:1)1 ................... 66
m
5.2.4 Representations Arising from the Orbit Class O(cid:1)2 ..... 66
r
5.2.5 Representations Arising from the Orbit Class O(cid:1)3 ..... 69
5.3 Symmetry Actions in 2+1 Dimensions .................... 69
5.3.1 Unitary Irreducible Representations of G ............ 69
6 Galilei Invariant Wave Equations ........................ 73
6.1 Wave Equations ........................................ 74
6.2 The 3+1 Dimensional Case.............................. 78
6.2.1 The Gyromagnetic Ratio........................... 83
6.3 The 2+1 Dimensional Case.............................. 84
6.4 Finite Dimensional Representations of the Euclidean Group .. 86
Appendix..................................................... 89
A.1 Dictionary of Mathematical Notions....................... 89
A.2 The Group of Automorphisms of a Hilbert Space............ 99
A.3 Induced Representation.................................. 100
References.................................................... 103
List of Frequently Occurring Symbols......................... 105
Index......................................................... 109
