Appendix a the Symmetry Representation Theorem

Home , Galilean transformation, Yesterday (time)

Appendix A Symrne try Rcpresentat ion Theorem 9 1 reference to symmetry principies, hecat~sewhatever one thinks the syrnrnelry group of nature map be, there is alwcsys another group whose colasequences are iden~icalexcept Jhr the rrbsence uf supersrlr~ctionrules.

Appendix A The Symmetry Representation Theorem

This appendix presents the proof of the fundamental theorem of wiper2 that any symmetry transformation can be represented on the Hilbert space of physical states by an operator that is either linear and unitary or antilinear and antiunitary. For our present purposes, the property of symmetry transformations on which we chiefly rely is that they are ray transformations T that preserve transition probabilities, in the sense that if and Y!2 are state-vectors belonging to rays 21 and 92then any state-vectors Yi and Y!; belonging to the transformed rays T.@]and T9f2 satisfy

We also require that a symmetry transformation should have an inverse that preserves transition probabilities in the same sense. To start, consider some complete orthonormal set of state-vectors YJk belonging to rags &, with and let YL be some arbitrary choice of statevectors belonging to the transformed rays T9k. From Eq. (2.A.11, we have

I{'Y;,T;)I~= ~(vkry!)~~ = ski. But (Yk,Y'k) is automatically real and positive, so this requires that it should have the value unity, and therefore it is easy to see that these transformed states VTi also form a complete set, for if there were any non-zero state-vector Yfthat was orthogonal to all of the Yb, then the inverse transform of the ray to which Yf belongs would consist of non-zero state-vectors Y" for which, for all k: which is impossible since the Vk were assumed to form a complete set. We must now establish a phase convention for the states YL. For this purpose, we single out one of the Yk,say Y1, and consider the state-vectors 92 2 Relativistic Quantum Mechalalcs belonging to some ray Yk,with k f 1. Any state-vector Ti belonging to the transformed ray TYkmay be expanded in the state-vectors Yi,

From Eq. (2.A.1) we have

1 and for 1 # k and l + 1 :

For any given k, by an appropriate choice of phase of the two state- vectors Yk and Yi we can clearly adjust the phases of the two non-zero coefficients ckk and ckl so that both coefficients are just 118. From now on, the state-vectors TL and Y; chosen in this way will be denoted UTk and UYk. As we have seen, 1 1 U-[vk + vI]= UYk = -[UTk + Uyl] . (2.A.5) Js $ However, it still remains to define UY for general state-vectors Y. Now consider an arbitrary state-vector Y! belonging to an arbitrary ray 9,and expand it in the Yk:

Any state Y" that belongs to the transformed ray TW may similarly be expanded in the complete orthonormal set UYk:

The equality of (Yk,y)12 and I(UYLk,yf)12 tells us that for all k (including k = 1):

1412= lcLl2, (2.A.8) while the equality of i(Yk, Y)1 and LIT^, Y!')12 tells us that for all k # 1:

The ratio of Eqs. (2.A.9)and (2.A.8) yields the formula which with Eq. (2.A.8) also requires Appendix A Symmetry Representation Theorem and therefore either or else

Furthermore, we can show that the same choice must be made for each k. (This step in the proof was omitted by Wigner.) To see this, suppose that for some k, we have Ck/C1= Ci/C;, while for some I # k, we have instead CI/CI = (C;/C;)*. Suppose also that both ratios are complex, so that these are really different cases. (This incidentally requires that k # 1 and I # 1, as well as k # I.) We will show that this is impossible. Define a state-vector (D [V1 Yk Yi]. Since all the ratios of the = ,. 3 + + coefficients in this state-vector are real, we must get the same ratios in any state-vector @' belonging to the transformed ray;

where a is a phase factor with lcll = 1. But then the equality of the transition probabilities I(@, CY)I and I(@',lyl)l requires that

and hence

This is only possible if

or, in other words, if

Hence either Ck/C1 or CI/C1 must be real for any pair k, 1, in contra- diction with our assumptions. We see then that for a given symmetry transformation T applied to a given state-vector CkYlk,we must have either Eq. (2.A.12) for all k, or else Eq. (2.A.13) for all k. Wigner ruled out the second possibility, Eq. (2.A.13), because as he showed any symmetry transformation for which this possibility is realized would have to involve a reversal in the time coordinate, and in the proof he presented he was considering only symmetries like rotations that do 94 2 Relativistic Quantum Mechanics not affect the direction of time. Here we are treating symmetries involving time-reversal on the same basis as all other symmetries, so we will have to consider that, for each symmetry T and state-vector CkCkVk, either Eq, (2.A.12) or Eq. (2.A.13) may apply. Depending on which of these alternatives is realized, we will now define UY to be the particular one of the state-vectors Y' belonging to the ray TB with phase chosen so that either C1 = C; or CI = c;', respectively. Then either

or else

It remains to be proved that for a given symmetry transformation, we must make the same choice between Eqs. (2.A.14) and (2.A.15) for arbitrary values of the coefficients Ck. Suppose that Eq. (2.A. 14) applies for a state-vector CkAkYk while Eq. (2.A.15) applies for a state-vector CkBkVk. Then the invariance of transition probabilities requires that

or equivalently 1m (A;Al)lm (B,'BI)= 0 . (2.A.16) .k I We cannot rule out the possibility that Eq. (2.A.14)may be satisfied for a pair of state-vectors CkAkYk and Ck&YE; belonging to different rays. However, for any pair of such state-vectors, with neither Ak nor Bk all of the same phase (so that Eqs. (2.A.14) and (2.A.15) are not the same), we can always find a third state-vector '& Ck Yk for which*

and also

* If Eor some pair k,l both A; A, and B; Br arc complex, then ct~onscall C s to vanish excepl Tor Ck and Cl,and chouse these twn coefficients to hsvc dineren1 phases. If Aidr is cumplex but B'BI is real for some pair k,l, then there must be some othcr pair m,n (possibly ahh either m or n bu~ nnl both equal to k or I) tor which BiB, ic complex. If ahAiA, is complex, then choose all Cs to vanish except for I;, and Cn, and choose these two coefficients Lo have diffcrcnt phase. If AkA, is rcal, Lhen choose all Cs to vanish except for CL,Cf ,C'm, and C,, and choose thcsc four coefficients all to have djfTercnt phases. The case whcrc BiB1 is complex but A;Al is red is handled in just the samc way. Appendix A Symmetry Representation Theorem 9 5

As we have seen, it follows from Eq. (2.A.17)that the same choice between Eqs. (2.A. 14) and (2.A.15) must be made for CkAkYk and CkCkYk, and it follows from Eq. (2.A.18) that the same choice between Eqs. (2.A.14) and (2.A.15) musl be made for CkBkYk and CkCkYk, SO the same choice between Eqs. (2.A.14) and (2.A.15) must also be made for the two state- vectors CkAkVk and CkBkYk with which we started. We have thus shown that for a given symmetry transformation T either a11 state-vectors satisfy Eq. (2.A.14) or else they all satisfy Eq. (2.A.15). It is now easy to show that as we have defined it, the quantum mechan- ical operator U is either linear and unitary or antilinear and antiunitary. First, suppose that Eq. (2.A.14) is satisfied for all state-vectors CkCkYk. Any two stale-vectors Y and @ may be expanded as

and so, using Eq. (2.A.141,

Using Eq. (2.A.14) again, this gives so U is linear. Also, using Eqs. (2.A.2) and (2.A.3), the scalar product of the transformed states is

and hence

so U is unitary. The case of a symmetry that satisfies Eq. (2.A.15) for all state-vectors may be dealt with in much the same way. The reader can probably supply the arguments without help, but since antilinear operators may be unfamiliar, we shall give the details here anyway. Suppose that Eq. (2.A-15) is satisfied for all state-vectors CkYlk.Any two state-vectors Y and @ may be expanded as before, and so: 96 2 Relativistic Qualatuna Mechanics

Using Eq. (2.A.15) again, this gives so U is antilinear. Also, using Eqs. (2.A.2) and (2.A.31, the scalar product of the transformed states is

and hence (UY, UO) = (Y!, a))',

Appendix B Group Operators and Homotopy Classes

In this appendix we shall prove the theorem stated in Section 2.7, that the phases of the operators U(T)for finite symmetry transformations T may be chosen so that these operators form a representation of the symmetry group, rather than a projective representation, provided (a) the generators of the group can be defined so that there are no central charges in the Lie algebra, and (b) the group is simply connected. We shall also comment on the projective representations encountered for groups that are not simply connected, and their relation to the homotopy classes of the group. To prove this theorem, let us recall the method by which we construct the operators corresponding to symmetry transformations. As described in Section 2.2, we introduce a set of real variables Oa to parameterize these transformations, in such a way that the transformations satisfy the composition rule (2.2.15):

We want to construct operators U(T(B))= U[O] that satisfy the corresponding conditionm

To do this, we lay down arbitrary 'standard' paths O$(s) in group parameter space, running from the origin to each point 8, with OE(0) = 0 and @;(I) = 8". and define Us[s)along each such path by the differential

Square brackets arc used here to distinguish U operators construckd as functions of the group parameters from those expressed as functions of Ihe group transformations themselves. 4. Galilean Invariance

4.1 Generators of infinitesimal transformations

The freedom of choice for reference frames includes more than rotations: one can displace the origin, translate it by a constant vector; or one can let that translation grow proportionally with time; the two frames are in relative motion at constant velocity. We'll consider only relative speeds that are small on the scale of the speed of light; see Problems 4-3 and 4-4 for other circumstances. Then time has an absolute significance (Galilean*-Newtonian relativity) apart from the freedom of displacing its origin. The infinitesimal transformations of these types are displayed by the space-time changes

t=t-8t, if = r -lir , with lir = + liw x r + liv t , (4.1.1) where 8t is a constant, as are the vectors liw, liv. The accompanying unitary operator is

u = 1 + iG ( 4.1.2) where, now

G = . P + liw . J + liv . N - lit H + litp , (4.1.3) and we want to recognize that we always have the freedom of a phase transformation. The names for the generators are derived from classical mechanics:

P: linear momentum vector, J: (already familiar) angular momentum vector, H: energy; Hamiltonian (or Hamiltont operator), N: no classical name, perhaps booster? But now we have to notice something. If we write U = 1 + iG, it is clear that G is dimensionless - it is given by pure numbers. But P, the product

*Galileo GALILEI (1564-1642) tSir William Rowan HAMILTON (1805-1865)

J. Schwinger, Quantum Mechanics © Springer-Verlag Berlin Heidelberg 2001 184 4. Galilean Invariance of length [L] by momentum [M L/T], or [M L2/T] - or equally well -8tH: time [T] times energy [M L2/T2], not to mention 8w .J: angle (dimensionless) times angular momentum [M L2/T) - has dimensions, those of action. It is clear that up to now we have been employing natural atomic units, not the arbitrary units of macroscopic physics. So, if we wish to use the latter, we must include a conversion factor:

(4.1.4) where fi, the unit of action, is (21f)-1 times Planck's* constant h. Experiment tells us that h fi = - = 1.05457 X 10-27 erg sec = 0.658212 e V fs (4.1.5) 21f

(1 eV = 1.602177 x 10-12 erg, electron-volt; 1 fs = 1O-15S, femto-second). It is important to recognize that the order in which these transformations, even infinitesimal ones, are made is important, in general. To use a familiar situation consider rotations. Compare 1,2:

r --+ r - 81w x r --+ r - 81w x r - 82 w x (r - 81w x r)

= r - 81 w x r - 82w x r + 82w x (81 W x r) (4.1.6) with 2,1:

(4.1.7)

The result of performing 1,2 and then the inverse of 2,1 is

r --+ r + 82w x (8 1 w x r) -81w x (82w x r) , ... =(hw x (r x b2W)

= r - (8 1w x 82 w) x r = r - 8[12JW x r , (4.1.8) i. e., another rotation described by

(4.1.9)

From the viewpoint of unitary transformations we are saying that U2 U1 i- UI U2 and

(4.1.10) which for infinitesimal transformations becomes

'Max Karl Ernst Ludwig PLANCK (1858-1947) 4.1 Generators of infinitesimal transformations 185

(4.1.12) since

U2 U1 = (1 + ) (1 + ) i 1 = 1 + fi(G 2 + Gd - li2G2G1 (4.1.13) and

(4.1.14)

And so we have

(4.1.15)

Now the only possibility for the scalar O[12Ji.p is a multiple of OlW' 02W, which is symmetrical in 1 and 2, not antisymmetrical. Hence o[12Ji.p = O. Then, written as 1 in [J, Ow . J] = Ow x J , (4.1.16) we recognize the characterization of a vector under rotations. This immediately tells us that the analogous considerations for the vectors P, N, and J will yield

[p, Ow . J] = Ow x P , 1 iii [N, Ow . J] = ow x N , (4.1.17) whereas, for the scalar H,

1 - [H Ow . J] = 0 . (4.1.18) iii ' How about translations? As

(4.1.19) 186 4. Galilean Invariance indicates, we have

(4.1.20) and 1 in [(h€· P, 82€· P] = M[12]

(4.1.22) or

and PxP=O. (4.1.23)

Similarly,

[Nk,Nd = 0, N x N = O. (4.1.24) But when we come to 1 in[8 . P,8v· N] = M

(dimension of M: mass) we can no longer conclude that 8

(4.1.26)

With regard to transformations that include time displacement, consider

t -+ t - 81 t -+ t - 81 t - 82t ,

r -+ r - 81vt -+ r - (hvt - (hv(t - 81t) , (4.1.27) so that (1,2) x (2,1)-1 leaves us with a net displacement

(4.1.28) which will have no counterpart in displacements or rotations. So 1 =8v8t·P+M

1 in[N,H] = -P, ( 4.1.30) 4.1 Generators of infinitesimal transformations 187 whereas 1 1 -[iSwin . J ' -6t H] = 0 and in-[is£.· P ' -6tH] = 0 (4.1.31) imply

[J,H] =0 and [P,H]=O. (4.1.32)

The commutators involving J are the response to rotations, distinguishing vectors and scalars. Now let's look at the P commutators, the response to translations. From the P equation in (4.1.17) we get 1 in [J, 6£.· P] = iSJ = is£. x P , (4.1.33) and since

(4.1.34) also

(4.1.35) and of course

( 4.1.36)

Both J and N show a response to translation which can be expressed by a vector R such that 1 is,,R = iii [R, is£. . P] = is£. ,

1 in [Rk' Pt] = iSkl . (4.1.37)

(4.1.38) and we write

J=RxP+S, (4.1.39) where the components of S commute with those of P,

(4.1.40) 188 4. Galilean Invariance

Since R is a vector we must have 1 in[R,8w. J] = 8w x R 1 = in [R, 8w x R· P + 8w . S] (4.1.41) which is certainly satisfied if

or R x R = 0 and [Rk, Bd = 0 . ( 4.1.42)

Also, since N generates a displacement proportional to t it must contain Pt, or

N=Pt-MR. (4.1.43)

In particular, for t = 0, N = - M R, and R x R = 0 follows from N x N = O. Inasmuch as Rand P are vectors, so is

L=RxP (4.1.44) and, in view of 1 1 in[L,8w. J] = in[L,8w. L] = 8w xL (4.1.45) one has

Lx L = inL (4.1.46) which implies that

S x S = inS. (4.1.47)

We see that

J=L+S (4.1.48) is the decomposition into external or orbital angular momentum L, and internal or spin angular momentum S. We have now recognized that the system as a whole is described by position vector R, momentum vector P, which for each direction in space con- stitute a q,p set of operators:

(4.1.49)

Accordingly all these operators have continuous spectra and have a classical limit. 4.1 Generators of infinitesimal transformations 189

Notice also that

R·L=R·RxP=RxR·P=O (4.1.50) which means that a rotation about the direction R has no effect, has zero quantum number,

c5 ( I = i ( Ic5w . L = 0 if c5w ex: R . ( 4.1.51)

But zero is an integer and therefore all possible values of I in L 2 ' = 1(1 + 1)!l,2 are integers,

1= 0,1,2, .... (4.1.52)

Now look at the information we have about H: 1 [J,H] =0, [P,H]=O, ifj,[N,H] = -P. (4.1.53)

The first says that H is a scalar, the second, according to

(R'lp = (4.1.54) 1 (R components are compatible) says

(4.1.55)

H does not depend on R; the third is 1 -in·[Pt - M R , H] = - P (4.1.56) or

(4.1.57) in' M But, according to (P components are compatible, too)

(4.1.58) we have

(4.1.59) or

p 2 H = 2M + Hint with V pHint= 0 . (4.1.60) 190 4. Galilean Invariance 4.2 Hamilton operator for a system of elementary particles

For us an elementary particle is defined as one without internal energy, or at least with inaccessible internal energy under the given circumstances. For atomic structure discussions the elementary particles are electrons and nuclei. For nuclear physics discussions, they are protons and neutrons, and so on. Let each elementary particle be described by independent variables r a, Pa, Sa and mass mao Then we construct P, J, N additively

P= LPa, a

J = L(ra X Pa + Sa) = R X P + S , a N= L(Pat-mara) =Pt-MR, (4.2.1) a where

M=Lma , (4.2.2) a a and indeed

= a b = L r;; = 8kl . (4.2.3) a =1 =dkl We write

L r a X Pa = L [R + (r a - R)] X Pa a a

= R X P + L (r a - R) X (Pa - P) , (4.2.4)

a ' y , internal variables since L ma (r a - R) = 0 and L (Pa - P) = 0 , (4.2.5) a a and get

S = L [(r a - R) X (Pa - P) + Sa] (4.2.6) a for the total internal angular momentum. Problems 191

If the constituents were isolated from each other we would have

(4.2.7)

More general we write

(4.2.8) with

(4.2.9) where V, the potential interaction energy, is a scalar function of the internal variables and the Sa and possibly others.

Problems

4-1 Verify explicitly that L = R x P obeys the angular momentum commutation relations (4.1.46). Can you think of a reason, based on the vector structure of L, for the fact that any component of Lin has only integer values?

4-2 Show that L·S commutes with J, L2, and S2. Then find the eigenvalues of L· s.

4-3 Einsteinian * relativity: Replace the first line in (4.1.1) by _ 1 1 t = t - -JED - 2Jv . r , c c where c is the speed of light, and the Galilean form is formally recovered in the limit c -+ 00 if (l/c)<5to -+ Jt is understood. Show that the commutators are the same, with two exceptions:

and

4-4 In consequence of these modified commutation relations, what needs to be altered in the equations introducing Rand S?

* Albert EINSTEIN (1879-1955) 192 4. Galilean Invariance

4-5 Photons have only spin angular momentum + 1 or -1 along their direction of motion. (Incidentally, helicity is a more fitting term than spin under these circumstances.) A light beam is deflected through the angle B. To what extent can you anticipate the dependence of the deflected beam's intensity on angle from the spin properties of a photon? [Hint: Recall Problem 3-5.] Chapter 9

Space-time symmetry transformations

In the last chapter, we set up a vector space which we will use to describe the state of a system of physical particles. In this chapter, we investigate the requirements of space-time symmetries that must be satisﬁed by a theory of matter. For particle velocities small compared to the velocity of light, the classical laws of nature, governing the dynamics and interactions of these particles, are invariant under the Galilean group of space-time transformations. It is natural to assume that quantum dynamics, describing the motion of non-relativistic particles, also should be invariant under Galilean transformations. Galilean transformation are those that relate events in two coordinate systems which are spatially rotated, translated, and time-displaced with respect to each other. The invariance of physical laws under Galilean transformations insure that no physical device can be constructed which can distinguish the di↵erence between these two coordinate systems. So we need to assure that this symmetry is built into a non-relativistic quantum theory of particles: we must be unable, by any measurement, to distinguish between these coordinate systems. More generally, a symmetry transformation is a change in state that does not change the results of possible experiments. We formulate this statment in the form of a relativity principle: Deﬁnition 16 (Relativity principle). If (⌃) represents the state of the system which refers to coordinate system ⌃, and if a(⌃) is the value of a| possiblei observable operator A(⌃) with eigenvector a(⌃) , also referring to system ⌃, then the probability of observing this measurement in coordinate system| ⌃i must Pa be the same as the probability a0 of observing this measurement in system ⌃0,where⌃0 is related to ⌃by a Galilean transformation. ThatP is, the relativity principle requires that:

2 2 0 = a(⌃0) (⌃0) = = a(⌃) (⌃) . (9.1) Pa |h | i| Pa |h | i| In quantum theory, transformations between coordinate systems are written in as operators acting on vectors in .Solet V (⌃0) = U(G) (⌃) , and a(⌃0) = U(G) a(⌃) , (9.2) | i | i | i | i where U(G) is the operator representing a Galilean transformation between ⌃0 and ⌃. Then a theorem by Wigner[1] states that: Theorem 16 (Wigner). Transformations between two rays in Hilbert space which preserve the same probabilities for experiments are either unitary and linear or anti-unitary and anti-linear. Proof. We can easily see that if U(G) is either unitary or anti-unitary, the statement is true. The reverse proof that this is the only solution is lengthy, and we refer to Weinberg [?][see Weinberg, Appendix A, p. 91] for a careful proof. The group of rotations and space and time translations which can be evolved from unity are linear unitary transformations. Space and time reversals are examples of anti-linear and anti-unitary transformations. We will deal with the anti-linear symmetries later on in this chapter.

83 9.1. GALILEAN TRANSFORMATIONS CHAPTER 9. SYMMETRIES

R v t X(t)

a X’(t’) F

F’

Figure 9.1: The Galilean transformation for Eq. (9.1).

We start this chapter by learning how to describe Galilean transformations in quantum mechanics, and how to classify vectors in Hilbert space according to the way they transform under Galilean transformations. In the process, we will obtain a description of matter, based on the irreducible representations of the Galilean group, and use this information to build models of interacting systems of particles and fields. The methods of finding unitary representations for the Galilean group in non-relativistic mechanics is similar to the same problem for the Poincarégroup in relativistic mechanics. The results for the Poincaré group are, perhaps, better known to physicists and well described in Weinberg[?, Chapter 2], for example. It turns out, however, that the group structure of the Galilean group is not not as simple as that of the Poincarégroup. The landmark paper by Bargmann[2] on unitary projective representations of continuous groups contains theorems and results which we use here. Ray representations of the Galilean group are also discusses by Hamermesh[?][p. 484]. We also use results from several papers by Levy-Leblond[3, 4, 5, 6] on the Galilei group. In the next section, we show that Galilean transformation form a group.

9.1 Galilean transformations

A Galilean transformation includes time and space translation, space rotations, and velocity boosts of the coordinate system. An “event” in a coordinate frame ⌃is given by the coordinates (x,t). The same event is described by the coordinates (x0,t0) in another frame ⌃0, which is rotated an amount R, displaced a distance a, moving at a velocity v, and using a clock running at a time t0 = t + ⌧, with respect to frame ⌃, as shown in Fig. 9.1. The relation between the events in ⌃and ⌃0 is given by the proper Galilean transformation:

x0 = R(x)+vt + a ,t0 = t + ⌧, (9.3) with R a proper real three-dimensional orthogonal matrix such that det R = +1. We regard the transformation (9.3) as a relationship between an event as viewed from two di↵erent coordinate frames. The basic premise of non-relativistic quantum mechanics of point particles is that it is impossible to distinguish between these two coordinate systems and so this space-time symmetry must be a property of the vector space which describes the physical system. We discuss improper transformations in Section 9.7. c 2009 John F. Dawson, all rights reserved. 84 CHAPTER 9. SYMMETRIES 9.1. GALILEAN TRANSFORMATIONS

9.1.1 The Galilean group We need to show that elements of a Galilean transformation form a group. We write the transformation as: ⌃0 = G(⌃), where ⌃refers to the coordinate system and G =(R, v, a,⌧)totheelementsdescribingthe transformation. A group of elements is deﬁned by the following four requirements: Deﬁnition 17 (group). A group is a set of objects, the elements of the group, which we call G, and a multiplication, or combination, ruleG for combining any two of them to form a product, subject to the following four conditions:

1. The product G1G2 of any two group elements must be another group element G3. 2. Group multiplication is associative: (G1G2)G3 = G1(G2G3). 3. There is a unique group element I, called the identity, such that IG= G for all G in the group. 1 1 1 4. For any G there is an inverse, written G such that GG = G G = I. We ﬁrst show that one Galilean transformation followed by a second Galilean transformation is also a Galilean transformation. This statement is contained in the following theorem: Theorem 17 (Composition rule). The multiplication law for the Galilean group is

G00 = G0G =(R0, v0, a0,⌧0)(R, v, a,⌧) , (9.4) =(R0R, v0 + R0v, a0 + R0a + v0⌧,⌧0 + ⌧) . Proof. We ﬁnd:

x0 = Rx + vt + a ,t0 = t + ⌧,

x00 = R0x0 + v0t0 + a0 = R0R x +(R0v + v0)t + R0a + v0⌧ + a0

R00x + v00t + a00 ⌘ t00 = t0 + ⌧ 0 = t + ⌧ + ⌧ 0 t + ⌧ 00 ⌘ where

R00 = R0R, v00 = R0v + v0

a00 = R0a + v0⌧ + a0 ⌧ 00 = ⌧ 0 + ⌧.

That is, R00 is also an orthogonal matrix with unit determinant, and v00 and a00 are vectors. Thus the Galilean group is the set of all elements G =(R, v, a,⌧), consisting of ten real parameters, three for the rotation matrixG R, three each for boosts v and for space translations a, and one for time translations ⌧. Deﬁnition 18. The identity element is 1 = (1, 0, 0, 0), and the inverse element of G is:

1 1 1 1 G =(R , R v, R (a v⌧), ⌧) , (9.5) as can be easily checked. Thus the elements of Galilean transformations form a group. Example 26 (Matrix representation). It is easy to show that the following 5 5 matrix representation of the Galilean group elements: ⇥ R va G = 01⌧ , (9.6) 0 0011 @ A forms a group, where group multiplication is deﬁned to be matrix multiplication: G00 = G0G. Here R is understood to be a 3 3 matrix and v and a are 3 1 column vectors. ⇥ ⇥ c 2009 John F. Dawson, all rights reserved. 85 9.1. GALILEAN TRANSFORMATIONS CHAPTER 9. SYMMETRIES

Remark 14. An inﬁnitesimal Galilean transformation of the coordinate system is given in vector notation by:

x =✓ x nˆ +v t +a , ⇥ (9.7) t =⌧.

The elements of the transformation are given by 1 + G,whereG =(✓, v, a, ⌧ ). Example 27. We can ﬁnd di↵erential representations of the generators of the transformation in classical physics. We start by considering complex functions (x,t) which transform “like scalars” under Galilean transformations, that is: 0(x0,t0)= (x,t) . (9.8) For inﬁnitesimal transformations, this reads:

0(x0,t0)= (x0 x,t0 t)= (x0,t0) x 0 (x0,t0) t@ (x0,t0)+ , (9.9) · r t0 ··· and, to ﬁrst order, the change in functional form of (x,t) is given by:

(x,t)= x +t@ (x,t) , (9.10) · r t

Here we have put x0 x and t0 t.Substituting( 9.7) into the above gives: ! ! (x,t)= ✓ nˆ x + t v +a +⌧@ (x,t) . (9.11) · ⇥ r · r · r t We deﬁne the ten di↵erential generator operators (J, K, P,H) of Galilean transformations by i (x,t)= ✓ nˆ J +v K a P +⌧H (x,t) , (9.12) ~ · · · Here we have introduced a constant ~ so as to make the units of J, K, P, and H to be the classical units of angular momentum, impulse, linear momentum, and energy, respectively.1 Comparing (9.11)to(9.12), we ﬁnd classical di↵erential representations of the generators:

~ ~t ~ @ J = x , K = , P = ,H= i~ . (9.13) i ⇥ r i r i r @t When acting on complex functions (x,t), these ten generators produce the corresponding changes in the functional form of the functions. Example 28. Using the di↵erential representation (9.13), it is easy to show that the generators obey the algebra:

[Ji,Jj]=i~ ✏ijkJk , [Ki,Kj]=0, [Ji,H]=0, [Ji,Kj]=i~ ✏ijkKk , [Pi,Pj]=0, [Pi,H]=0, (9.14) [Ji,Pj]=i~ ✏ijkPk , [Ki,Pj]=0, [Ki,H]=i~ Pi .

9.1.2 Group structure If the generators of a group all commute, then the group is called Abelian. An invariant Abelian subgroup consists of a subset of generators that commute with each other and whose commutators with any other member of the group also belong to the subgroup. For the Galilean group, the largest Abelian subgroup is the six-parameter group =[L, P] generating boosts and translations. The largest abelian subgroup of the factor group, / , is theU group =[H], generating time translations. This leaves the semi-simple group G U D 1 The size of ~ is ﬁxed by the physics. c 2009 John F. Dawson, all rights reserved. 86 CHAPTER 9. SYMMETRIES 9.2. GALILEAN TRANSFORMATIONS

=[J], generating rotations. A semi-simple group is one which transform among themselves and cannot be reducedR further by removal of an Abelian subgroup. So the Galilean group can be written as the semidirect product of a six parameter abelian group with the semidirect product of a one parameter abelian group by a three parameter simple group , U D R =( ) . (9.15) G R⇥D ⇥U In contrast, the Poincar´egroup is the simidirect product of a simple group generating Lorentz transformations by an abelian group generating space and time translations, L C = . (9.16) P L⇥C 9.2 Galilean transformations in quantum mechanics

Now let (⌃) be a vector in which refers to a speciﬁc coordinate system ⌃and let (⌃0) be a vector | i V | i which refers to the coordinate system ⌃0 = G⌃. Then we know by Wigner’s theorem that:

(⌃0) = U(G) (⌃) , (9.17) | i | i where U(G) is unitary.2 In non-relativistic quantum mechanics, we want to ﬁnd unitary transformations U(G) for the Galilean group. We do this by applying the classical group multiplication properties to unitary transformations. That is, if (9.17) represents a transformation from ⌃to ⌃0 by G, and a similar relation holds for a transformation from ⌃0 to ⌃00 by G0, then the combined transformation is given by:

(⌃00) = U(G0) (⌃0) = U(G0) U(G) (⌃) . (9.18) | i | i | i

However the direct transformation from ⌃to ⌃00 is given classically by G00 = G0G, and quantum mechanically by: (⌃00) 0 = U(G00) (⌃) = U(G0G) (⌃) . (9.19) | i | i | i Now (⌃00) and (⌃00) 0 must belong to the same ray, and therefore can only di↵er by a phase. Thus we can deduce| that:i | i i(G0,G)/ U(G0) U(G)=e ~ U(G0G) , (9.20) where (G0,G) is real and depends only on the group elements G and G0. Unitary representations of operators which obey Eq. (9.20) with non-zero phases are called projective representations. If the phase (G0,G) = 0, they are called faithful representations. The Galilean group generally is projective, not faithful.3 The group composition rule, Eq. (9.20), will be used to ﬁnd the unitary transformation U(G). Now we can take the unit element to be: U(1) = 1. So using the group composition rule (9.20), unitarity requires that:

1 1 1 i(G ,G)/ U †(G)U(G)=U (G)U(G)=U(G )U(G)=e ~ U(1, 0) = 1 . (9.21)

1 so that (G ,G) = 0. We will use this unitarity requirement in section 9.2.1 below. Infinitesimal transformations are generated from the unity element by the set G =(!,v, a, ⌧), where !ij = ✏ijknk✓ = !ji is an antisymmetric matrix. We write the unitary transformation for this infinitesimal transformation as: i U(1 + G)=1+ !ij Jij/2+vi Ki ai Pi +⌧H + ~ ··· (9.22) i n o =1+ ✓ nˆ J +v K a P +⌧H + , ~ · · · ··· n o 2We will consider anti-unitary symmetry transformations later. 3In contrast, the Poincarégroup is faithful. c 2009 John F. Dawson, all rights reserved. 87 9.2. GALILEAN TRANSFORMATIONS CHAPTER 9. SYMMETRIES where J , K , P and H are operators on which generater rotations, boosts, and space and time translations, i i i V respectively. Here !ij = ✏ijk nk ✓ is an antisymmetric matrix representing an infinitesimal rotation about an axis defined by the unit vector nk by an angle ✓. In a similar way, we write the antisymmetric matrix of operators Jij as Jij = ✏ijkJk,whereJk is a set of three operators. Remark 15. Again, we have introduced a constant ~ so that the units of the operators J, K, P, and H are given by units of angular momentum, impulse, linear momentum, and energy, respectively. The value of ~ must be fixed by experiment.4

Remark 16. The sign of the operators Pi and H, relative to Jk in (9.22) is arbitrary — the one we have chosen is conventional.

In the next section, we ﬁnd the phase factor (G0; G)inEq.(9.20) for unitary representations of the Galilean group.

9.2.1 Phase factors for the Galilean group.

1 The phases (G0,G) must obey basic properties required by the transformation rules. Since U (G)U(G)= 1 U(G )U(G) = 1, we ﬁnd from the unitarity requirement (9.21),

1 (G ,G)=0. (9.23)

Also, the associative law for group transformations,

U(G00)(U(G0)U(G)) = (U(G00)U(G0)) U(G) , requires that

(G00,G0G)+(G0,G)=(G00,G0)+(G00G0,G) . (9.24)

From (9.23) and (9.24), we easily obtain (1, 1) = (1,G)=(G, 1) = 0. Eqs. (9.23) and (9.24) are the defining equations for the phase factor (G0,G), and will be used in Bargmann’s theorem (18)tofindthe phase factor below. Note that (9.23) and (9.24) can be satisfied by any (G0,G) of the form

(G0,G)=(G0G) (G0) (G) . (9.25) Then the phase can be eliminated by a trivial change of phase of the unitary transformation, U¯(G)= i(G) e U(G). Thus two phases (G0,G) and 0(G0,G) which di↵er from each other by functions of the form (9.25) are equivalent. For Galilean transformations, unlike the case for the Poincarégroup, the phase (G0,G) cannot be eliminated by a simple redefinition of the unitary operators. This phase makes the stude of unitary representations of the Galilean group much harder than the Poincarégroup in relativistic quantum mechanics. It turns out that the phase factors for the Galilean group are not easy to find. The result is stated in a theorem due to Bargmann[2]:

Theorem 18 (Bargmann). The phase factor for the Galilean group is given by:

M (G0,G)= v0 R0(a) v0 R0(v) ⌧ a0 R0(v) , (9.26) 2 { · · · } with M any real number.

4 Plank introduced ~ in order to make the classical partition function dimensionless. The value of ~ was ﬁxed by the experimental black-body radiation law. c 2009 John F. Dawson, all rights reserved. 88 CHAPTER 9. SYMMETRIES 9.2. GALILEAN TRANSFORMATIONS

Proof. A proper Galilean transformation is given by Eq. (9.3). The group multiplication rules are given in Eq. (9.4):

R00 = R0R,

v00 = v0 + R0(v) , (9.27) a00 = a0 + v0⌧ + R0(a) ,

⌧ 00 = ⌧ 0 + ⌧.

We ﬁrst note that v and a transform linearly. Therefore, it is useful to introduce a six-component column matrix ⇠ and a 6 6 matrix ⇥(⌧), which we write as: ⇥ v 10 ⇠ = , ⇥(⌧)= , (9.28) a ⌧ 1 ✓ ◆ ✓ ◆ so that we can write the group multiplication rules for these parameters as:

⇠00 =⇥(⌧) ⇠0 + R0 ⇠, (9.29) which is linear in the ⇠ variables. We label the rest of the parameters by g =(R,⌧), which obey the group multiplication rules: R00 = R0R,⌧00 = ⌧ 0 + ⌧. (9.30) We note here that the unit element of g is g =(1, 0). We also note that the matrices ⇥(⌧) are a faithful representation of the subgroup of ⌧ transformations. That is, we ﬁnd:

⇥(⌧ 00)=⇥(⌧ 0)⇥(⌧) . (9.31)

We seek now the form of (G0,G) by solving the deﬁning equation (9.24):

(G00,G0G)+(G0,G)=(G00,G0)+(G00G0,G) . (9.32)

The only way this can be satisﬁed is if (G0,G) is bilinear in ⇠, because the transformation of these variables is linear. Thus we make the Ansatz: T (G0,G)=⇠0 (g0,g) ⇠, (9.33) where (g0,g)isa6 6 matrix, but depends only on the elements g and g0. We now work out all four terms in Eq. (9.32). We ﬁnd:⇥

T (G00,G0G)=⇠00 (g00,g0g) ⇥(⌧) ⇠0 + R0 ⇠ T T = ⇠00 (g00,g0g)⇥ (⌧) ⇠0 + ⇠00 ⇤(g00,gg) R0 ⇠, T (G0,G)=⇠0 (g0,g) ⇠, (9.34) T (G00,G0)=⇠00 (g00,g0) ⇠0 , T T T T (G00G0,G)= ⇠0 R00 + ⇠00 ⇥ (⌧ 0) (g00g0,g) ⇠ T T T T = ⇥⇠0 R00 (g00g0,g) ⇠ + ⇠⇤00 ⇥ (⌧ 0)(g00g0,g) ⇠. Substituting these results into (9.32), and equating coecients for the three bilinear forms, we ﬁnd for the three pairs: (⇠0; ⇠), (⇠00; ⇠0), and (⇠00; ⇠):

T (g0,g)=R00 (g00g0,g) , (9.35)

These relations provide functional equations for the matrix elements. We start by using the orthogonality of R and writing (9.35) in the form: (g00g0,g)=R00 (g0,g) (9.38)

Since g0 is arbitrary, we can set it equal the unit element: g0 =(1, 0). Then g00g0 = g00, and we ﬁnd:

(g00,g)=R00 (1,g) . (9.39)

When this result is substituted into (9.36) and (9.37), we ﬁnd:

R00 (1,g0g)⇥(⌧)=R00 (1,g0) (9.40) T R00 (1,g0g) R0 =⇥ (⌧ 0) R00R0 (1,g) . (9.41) and from (9.40), we ﬁnd: (1,g0g)⇥(⌧)=(1,g0) . (9.42)

Here g0 is arbitrary, so that we can it to the unit element: g0 = 1, and ﬁnd:

(1,g)⇥(⌧)=(1, 1) . (9.43)

Now in (9.41), R00 and R0 act only on vectors and commute with the matrices ⇥and , so we can write this as: T (1,g0g)=⇥ (⌧ 0)(1,g) . (9.44) Again in (9.44), we can set g = 1, from which we ﬁnd:

T (1,g0)=⇥ (⌧ 0)(1, 1) . (9.45)

So combining (9.43) and (9.45), we ﬁnd that (1, 1) must satisfy the equation:

(1, 1) = ⇥T (⌧)(1, 1) ⇥(⌧) , (9.46) for all values of ⌧. Which means that (1, 1) must be a constant 6 6 matrix, independent of ⌧. In order to solve (9.46), we write out (1, 1) in component form: ⇥

(1, 1) = 11 12 , (9.47) ✓ 21 22◆ so that (9.46)requires:

2 11 =11 + ⌧ (12 +21)+⌧ 22 , (9.48)

12 =12 + ⌧ 22 , (9.49)

21 =21 + ⌧ 22 , (9.50)

22 =22 , (9.51) which must hold for all values of ⌧. This is possible only if = 0, and that = . is then 22 21 12 11 arbitrary. So let us put 12 = M/2 and 11 = M 0/2. So the general solution for the phase matrix contains two constants. We write the result as:

M M 0 01 10 (1, 1) = Z + Z0 , where Z = ,Z0 = , (9.52) 2 2 10 00 ✓ ◆ ✓ ◆ From Eqs. (9.33), (9.39), and (9.45), we ﬁnd:

Recall that R0 commutes with ⇥(⌧) and (1, 1). It turns out that the term involving M 0Z0 is a trivial phase. For this term, we ﬁnd:

M 0 T T Z0 (G0,G)= ⇠0 ⇥ (⌧) Z0 R0(⇠) 2 (9.54) M 0 M 0 2 2 2 = v0 R0(v)= v00 v v0 , 2 · 4 So (9.54) is a trivial phase and can be absorbed into the deﬁnition of U(g). So then from Eq. (9.52), the phase is given by:

M T T M T (G0,G)=+ ⇠0 ⇥ (⌧) ZR0(⇠)= [ R0(⇠)] Z ⇥(⌧) ⇠0 . 2 2 (9.55) M = v0 R0(a) v0 R0(v) ⌧ a0 R0(v) , 2 { · · · } which is what we quoted in the theorem. In the ﬁrst line, we have used the fact that Z is antisymmetric: ZT = Z. This phase is non-trival! For example, we might try to do the same tricks we used for the trival phase in Eq. (9.54), and write:

T T T T ⇠00 Z⇠00 = [ R0(⇠)] + ⇠0 ⇥ (⌧) Z ⇥(⌧) ⇠0 + R0(⇠) (9.56) T T T T T = ⇠0 Z⇠0 + ⇠ Z⇠+ ⇠0 ⇥ (⌧) ZR0(⇠)+[R0( ⇠)] Z ⇥(⌧) ⇠0 . But the last two terms cancel rather than add because of the antisymmetry of Z. So we cannot turn (9.55) into a trival phase the way we did for (9.54). This completes the proof. Remark 17. Bargmann gave this phase in his classic paper on continuous groups[2], and indicated how he found it in a footnote to that paper. Notice that M appears here as an undetermined multiplicative parameter. Since we have introduced a constant ~ with the dimensions of action in the deﬁnition of the phase, M has units of mass. We can write the phase as:

1 (G0,G)= MR0 v0a a0 v v0v ⌧] (9.57) 2 ij{ i j i j i j 1 Notice that (G ,G) = 0. The phase for inﬁnitesimal transformations are given by: (G, 1+G)= 1 MR [v a a v ]+ , (9.58) 2 ij i j i j ··· (1 + G, G)= 1 M [v (a v ⌧) a v ]+ , 2 i i i i i ··· Next, we ﬁnd the transformation properties of the generators.

9.2.2 Unitary transformations of the generators In this section, we ﬁnd the unitary transformation U(G) for the generators of the Galilean group. We start by ﬁnding the transformation rules for all the generators. This is stated in the following theorem: Theorem 19. The generators transform according to the rules:

U †(G) J U(G)=R J + K v¯ + a¯ (P + M v¯) , (9.59) { ⇥ ⇥ } U †(G) K U(G)=R K (P + Mv¯) ⌧ + Ma¯ , (9.60) { } U †(G) P U(G)=R P + Mv¯ , (9.61) { } 1 2 U †(G) HU(G)=H + v¯ P + Mv . (9.62) · 2 1 1 where v¯ = R (v) and a¯ = R (a). c 2009 John F. Dawson, all rights reserved. 91 9.2. GALILEAN TRANSFORMATIONS CHAPTER 9. SYMMETRIES

Proof. We start by considering the transformations:

U †(G) U(1 + G) U(G) , (9.63) where G and 1 + G are two di↵erent transformations. On one hand, using the deﬁnition (9.22) for inﬁnitesimal transformations in terms of the generators, (9.63) is given by: i i 1+ !ij U †(G) Jij U(G)+ vi U †(G) Ki U(G) (9.64) 2~ ~ i i ai U †(G) Pi U(G)+ ⌧U†(G) HU(G)+ ~ ~ ··· On the other, using the composition rule (9.20), Eq. (9.63) can be written as:

1 i[(G ,(1+G)G)+((1+G),G)]/ 1 e ~ U(G (1 + G)G) (9.65) i(G,G)/ = e ~ U(1 + G0) .

1 where G0 = G GG. Working out this transformation, we ﬁnd the result:

!ij0 = RkiRlj !kl ,

vi0 = Rji (!jk vk +vj) , a0 = R (! a +v ⌧ +a v ⌧) i ji jk k j j j ⌧ 0 =⌧, and the phase (G, G)isdeﬁnedby:

1 (G, G)=(G , (1 + G)G)+(1 + G, G) . (9.66)

We can simplify the calculation of the phase using an identity derived from (9.24):

1 1 (G, G (1 + G)G)+(G , (1 + G)G) 1 1 = (G, G )+(GG , (1 + G)G)=(1, (1 + G)G)=0,

1 and therefore, since G (1 + G)G =1+G0,wehave:

1 (G , (1 + G)G)= (G, 1+G0) . So the phase (G, G) is given by:

(G, G)=(1 + G, G) (G, 1+G0) . (9.67) Now using (9.58), we ﬁnd to ﬁrst order:

(1 + G, G)= 1 M [v (a v ⌧) a v ]+ , 2 i i i i i ··· 1 (G, 1+G0)= MR [v a0 a v0 ]+ , 2 ij i j i j ··· = 1 M v (! a +v ⌧ +a v2⌧) a (v +! v ) 2 { i ij j i i i i ij j } + , ··· from which we ﬁnd,

For the unitary operator U(1 + G0), we find: i i i i U(1 + G0)=1+ !ij0 Jij + vi0 Ki ai0 Pi + ⌧ 0 H + , 2~ ~ ~ ~ ··· i =1+ !ij [RikRjlJkl +2Ril(vjKl ajPl)] 2~ i i + vi Rij(Kj ⌧Pj) ai RijPj ~ ~ i + ⌧ (H + RijviPj)+ , (9.69) ~ ··· Combining relations (9.68) and (9.69), we find, to first order, the expansion: i(G,G)/ e ~ U(1 + G0) i =1+ !ij [RikRjlJkl +2Ril(vjKl ajPl)+M(aivj ajvi)] 2~ i + vi [Rij(Kj ⌧Pj)+M(ai vi⌧)] ~ i ai [RijPj + Mvi] ~ i 1 2 + ⌧ [H + RijviPj + Mv ]+ , (9.70) ~ 2 ···

Comparing coecients of !ij,vi,ai, and ⌧ in (9.64) and (9.70), we get:

U †(G) J U(G)=R R J +2R (v K a P )+M(a v a v ) ij ik jl kl il j l j l i j j i = R R J +(K0v K0 v ) (P 0a P 0a )+M(a v a v ) ik jl kl i j j i i j j i i j j i U †(G) K U(G)=R (K ⌧P )+M(a v ⌧) i ij j j i i U †(G) Pi U(G)=RijPj + Mvi 1 2 U †(G) HU(G)=H + viPi0 + 2 Mv where, Ki0 = RijKj and Pi0 = RijPj. In the second line, we have used the antisymmetry of Jij.These equations simplify if we rewrite them in terms of the components of the angular momentum vector Jk rather than the antisymmetric tensor Jij. We have the deﬁnitions:

Jij = ✏ijkJk ,

K0v K0 v = ✏ [K0 v] , i j j i ijk ⇥ k P 0a P 0a = ✏ [P0 a] , i j j i ijk ⇥ k v a v a = ✏ [v a] . i j j i ijk ⇥ k The identity, RikRjl ✏klm =det[R ] ✏ijnRnm , (9.71) is obtained from the deﬁnition of the determinant of R and the orthogonality relations for R. For proper transformations, which is what we consider here, det[ R ] = 1. So the above equations become, in vector notation,

U †(G) J U(G)=R J + K v¯ + a¯ (P + M v¯) , { ⇥ ⇥ } U †(G) K U(G)=R K (P + Mv¯) ⌧ + Ma¯ , { } U †(G) P U(G)=R P + Mv¯ , { } 1 2 U †(G) HU(G)=H + v¯ P + Mv . · 2 1 1 where ¯v = R (v) and ¯a = R (a). This completes the proof of the theorem, as stated. c 2009 John F. Dawson, all rights reserved. 93 9.2. GALILEAN TRANSFORMATIONS CHAPTER 9. SYMMETRIES

Exercise 8. Using the indentity (9.71)withdet[R ] = +1, show that R(A B)=R(A) R(B). ⇥ ⇥ We next turn to a discussion of the commutation relations for the generators.

9.2.3 Commutation relations of the generators In this section, we prove a theorem which gives the commutation relations for the generators of the Galilean group. The set of commutation relations for the group can be thought of as rules for “multiplying” any two operators, and are called a Lie algebra.

Theorem 20. The ten generators of the Galilean transformation satisfy the commutation relations:

[Ji,Jj]=i~ ✏ijkJk , [Ki,Kj]=0, [Ji,H]=0, [Ji,Kj]=i~ ✏ijkKk , [Pi,Pj]=0, [Pi,H]=0, (9.72) [Ji,Pj]=i~ ✏ijkPk , [Ki,Pj]=i~ Mij , [Ki,H]=i~ Pi .

Proof. The proof starts by taking each of the transformations U(G) in theorem 19 to be infinitesimal. These infinitesimal transformations have nothing to do with the infinitesimal transformations in the previous theorem — they are di↵erent transformations. We start with Eq. (9.59) where we find, to first order:

i i i i 1 J ✓ K v + P a H⌧ + k k k k k k ··· ⇢ ~ ~ ~ ~ i i i i J 1+ J ✓ + K v P a + H⌧ + ⇥ i k k k k k k ··· ⇢ ~ ~ ~ ~ = J + ✏ J ✓ + ✏ K v + ✏ a P + . i ijk j k ijk j k kji k j ···

Comparing coecients of ✓k,vk,ak, and ⌧, we ﬁnd the commutators of Ji with all the other generators:

[Ji,Jj]=i~ ✏ijkJk , [Ji,Pj]=i~ ✏ijkPk , [Ji,Kj]=i~ ✏ijkKk , [Ji,H]=0.

From (9.60), we ﬁnd, to ﬁrst order:

i i i i 1 J ✓ K v + P a H⌧ + k k k k k k ··· ⇢ ~ ~ ~ ~ i i i i K 1+ J ✓ + K v P a + H⌧ + ⇥ i k k k k k k ··· ⇢ ~ ~ ~ ~ = K + ✏ K ✓ + Ma P ⌧ + , i ijk j k i i ··· from which we ﬁnd the commutators of Ki will all the generators. In addition to the ones found above, we get: [Ki,Kj]=0, [Ki,Pj]=i~ Mij , [Ki,H]=i~ Pi .

The commutators of Pi with the generators are found from (9.61). We ﬁnd, to ﬁrst order:

i i i i 1 J ✓ K v + P a H⌧ + k k k k k k ··· ⇢ ~ ~ ~ ~ i i i i P 1+ J ✓ + K v P a + H⌧ + ⇥ i k k k k k k ··· ⇢ ~ ~ ~ ~ = P + ✏ P ✓ + Mv + , i ijk j k i ··· c 2009 John F. Dawson, all rights reserved. 94 CHAPTER 9. SYMMETRIES 9.2. GALILEAN TRANSFORMATIONS

from which we ﬁnd the commutators of Ki with all the generators. In addition to the ones found above, we get:

[Pi,Pj]=0, [Pi,H]=0. The last commmutation relations of H with the generators conﬁrm the previous results. This completes the proof.

The phase parameter M is called a central charge of the Galilean algebra.

9.2.4 Center of mass operator For M = 0, it is useful to define operators which describes the location and velocity of the center of mass: 6 Definition 19. The center of mass operator X is defined at t =0byX = K/M . We also define the velocity of the center of mass as V = P/M .

If no external forces act on the system, the center of mass changes in time according to:

X(t)=X + V t. (9.73)

There can still be internal forces acting on various parts of the system: we only assume here that the center of mass of the system as a whole moves force free. Using the transformation rules from Theorem 19, X(t) transforms according to:

U †(G) X(t0) U(G)=U †(G) K + P (t + ⌧) U(G)/M { } = R K (P + Mv¯) ⌧ + Ma¯ + P (t + ⌧)+Mv¯ (t + ⌧) /M { } (9.74) = R K + P t /M + v t + a { } = RX(t)+v t + a , where t0 = t + ⌧.

Di↵erentiating (9.74)withrespecttot0,weﬁnd:

U †(G) X˙ (t0) U(G)=RX˙ (t)+v ,

U †(G) X¨ (t0) U(G)=RX¨ (t) , so the acceleration of the center of mass is an invariant. We can rewrite the transformation rules and commutation relations of the generators of the Galilean group using X = K/M and V = P/M rather than K and P. From Eqs. (9.59–9.62), we ﬁnd:

U †(G) J U(G)=R J + MX v¯ + M a¯ (V + v¯) { ⇥ ⇥ } = R J + M (X + a¯) v¯ + M a¯ V , { ⇥ ⇥ } U †(G) X U(G)=R X (V + v¯) ⌧ + a¯ , (9.75) { } U †(G) V U(G)=R V + v¯ , { } 1 2 U †(G) HU(G)=H + Mv¯ V + Mv . · 2 1 1 where v¯ = R (v) and a¯ = R (a). Eqs. (9.72) become:

[Ji,Jj]=i~ ✏ijkJk , [Xi,Xj]=0, [Ji,H]=0, [Ji,Xj]=i~ ✏ijkXk , [Pi,Pj]=0, [Pi,H]=0, (9.76) [Ji,Pj]=i~ ✏ijkPk , [Xi,Pj]=i~ ij , [Xi,H]=i~ Vi . c 2009 John F. Dawson, all rights reserved. 95 9.2. GALILEAN TRANSFORMATIONS CHAPTER 9. SYMMETRIES

Remark 18. For a single particle, the center of mass operator is the operator which describes the location of the particle. The existence of such an operator means that we can localize a particle with a measurement of X. The commutation relations between X and the other generators are as we might expect from the canonical quantization postulates which we study in the next chapter. Here, we have obtained these quantization rules directly from the generators of the Galilean group, and from our point of view, they are consequences of requiring Galilean symmetry for the particle system, and are not additional postulates of quantum theory. We shall see in a subsequent chapter how to construct quantum mechanics from classical actions. Remark 19. Since in the Cartesian system of coordinates, X and P are Hermitian operators, we can always write an eigenvalue equation for them:

X x = x x , (9.77) | i | i P p = p p , (9.78) | i | i where xi and pi are real continuous numbers in the range

9.2.5 Casimir invariants Casimir operators are operators that are invariant under the transformation group and commute with all the generators of the group. The Galilean transformation is rank two, so we know from a general theorem in group theory that there are just two Casimir operators. These will turn out to be what we will call the internal energy W and the magnitude of the spin S, or internal angular momentum. We start with the internal energy operator. Definition 20 (Internal energy). For M = 0, we define the internal energy operator W by: 6 P 2 W = H . (9.79) 2M Theorem 21. The internal energy, defined Eq. (9.79), is invariant under Galilean transformations: Proof. Using Theorem 19,wehave:

2 1 2 [R P + M ¯v ] U †(G) WU(G)=H + ¯v P + Mv { } · 2 2M P 2 = H = W, 2M as required. The internal energy operator W is Hermitian and commutes with all the group generators, its eigenvalues w can be any real number. So we can write:

P 2 H = W + . (9.80) 2M The orbital and spin angular momentum operators are defined by: Definition 21 (Orbital angular momentum). For M = 0, we define the orbital angular momentum by: 6 L = X P =(K P)/M . (9.81) ⇥ ⇥ The orbital angular momentum of the system is independent of time:

Definition 22 (Spin). For M = 0, we define the spin, or internal angular momentum by: 6 S = J L , (9.83) where L is defined in Eq. (9.81). The spin is what is left over after subracting the orbital angular momentum from the total angular momentum. Since the orbital angular momentum is not defined for M = 0, the same is true for the spin operator. However for M = 0, we can write: 6 J = L + S . (9.84) The following theorem describes the transformation properties of the orbital and spin operators. Theorem 22. The orbital and spin operators transform under Galilean transformations according to the rule:

U †(G) L U(G)=R L + X ( Mv¯ )+(a¯ ( V + v¯ ) ⌧ ) P , (9.85) { ⇥ ⇥ } U †(G) S U(G)=R S , (9.86) { } and obeys the commutation relations:

[ Li,Lj ]=i~ ✏ijkLk , [ Si,Sj ]=i~ ✏ijkSk , [ Li,Sj ]=0. (9.87) Proof. The orbital results are easy to prove using results from Eqs. (9.75). For the spin, using theorem 19, we ﬁnd:

U †(G) S U(G)=R J + K ¯v + ¯a (P + M ¯v ) { ⇥ ⇥ } R K (P + M ¯v ) ⌧ + M ¯a R P + M ¯v /M { }⇥ { } = R J + K ¯v + ¯a (P + M ¯v ) { ⇥ ⇥ [ K (P + M ¯v ) ⌧ + M ¯a ] [ P + M ¯v ]/M ⇥ } = R J (K P)/M = R S , { ⇥ } { } as required. The commutator [ Li,Jj ] = 0 is easy to establish. For [ Li,Lj ], we note that:

[ Li,Lj ]=✏inm✏jn0m0 [ XnPm,Xn0 Pm0 ]

= ✏inm✏jn0m0 Xn0 [ Xn,Pm0 ] Pm + Xn [ Pm,Xn0 ] Pm0

= i~ ✏inm✏jn m n,m Xn Pm n ,m Xn Pm 0 0 0 0 0 0 = i~ ✏inm✏jn n Xn Pm ✏inm✏jmm Xn Pm (9.88) 0 0 0 0 = i~ ( mjin mn ij ) Xn Pm ( im nj ijnm ) Xn Pm 0 0 0 0 0 0 = i~ Xi Pj ij ( Xm Pm ) Xj Pi + ij ( Xn Pn ) = i~ Xi Pj Xj Pi = i~ ✏ijk Lk , as required. The last commutator [ Si,Sj ] follows directly from the commutator results for Ji and Li.

Remark 20. Additionally, we note that [ Si,Xj ]=[Si,Pj ]=[Si,H] = 0. Remark 21. So this theorem showns that even under boosts and translations, in addition to rotations, the spin operator is sensitive only to the rotation of the coordinate system, which is not true for either the orbital angular momentum or the total angular momentum operators. However the square of the spin vector operator S2, is invariant under general Galilean transformations,

1 2 2 U (G) S U(G)=S , (9.89) and is the second Casimir invariant. In Section ??, we will ﬁnd that the possible eigenvalues of S2 are given by: s =0, 1/2, 1, 3/2, 2,.... c 2009 John F. Dawson, all rights reserved. 97 9.2. GALILEAN TRANSFORMATIONS CHAPTER 9. SYMMETRIES

Remark 22. To summerize this section, we have found two hermitian Casimir operators, W and S2,which are invariant under the group . We can therefore label the irreducible representations of by the set of quantities: [M w, s], where w andG s label the eigenvalues of these operators, and M the centralG charge. So we can ﬁnd| common eigenvectors of W , S2, and either X or P. We write these as:

[M w, s]; x, , and [M w, s]; p, . (9.90) | | i | | i Here labels the component of spin. The latter eigenvector is also an eigenvector of H, with eigenvalue:

p2 H [M w, s]; p, = E [M w, s]; p, ,E= w + . (9.91) | | i w,p | | i w,p 2M We discuss the massless case in Section 9.2.8.

9.2.6 Extension of the Galilean group If we wish, we may extend the Galilean group by considering M to be a generator of the group. This is because the phase factor (G0,G) is linear in M and M commutes with all elements of the group. Thus we can invent a new group element ⌘ and write:

G˜ =(G,⌘)=(R, v, a,⌧,⌘) , (9.92) and which transforms according to the rule:

G˜0G˜ =(G0G,⌘0 + ⌘ + ⇠(G0,G)) , (9.93) where ⇠(G0,G) is the coecient of M in (9.26) 1 ⇠(G0,G)= v0 R0 v ⌧ + a0 R0 v v0 R0 a . (9.94) 2{ · · · } The infinitesimal unitary operators in Hilbert space become: i U˜(1 + G˜)=1+ J nˆ ✓ + K v P a + H⌧ + M⌘ + , (9.95) ~ { · · · } ··· and since M is now regarded as a generator and ⌘ as a group element, the extended eleven parameter Galilean group can now be represented as a true unitary representation rather than a projective representation: the phase factor has been redefined as a transformation property of the extended group element ⌘, and the phase M redefined as a operator. For the extended Galilean group ˜ with M = 0, the largest abelian invariant subgroup is now the five dimensional subgroup ˜ =[P,H,M]G generating6 space and time translations plus ⌘. The abelian invariant subgroup of the factorC group ˜/ ˜ is then the three parameter subgroup =[K] generating boosts, leaving the semi-simple three-dimensionalG C group of rotations =[R]. So theV extended Galilean group has the product structure: R ˜ =( ) ˜. (9.96) G R⇥V ⇥ C Here the subgroup =[J, K] generates the six dimensional group of rotations and boosts. R⇥V 9.2.7 Finite dimensional representations We examine in this section finite dimensional representations of the subgroup =[J, K] of rotations and boosts. These generators obey the subalgebra: R⇥V

In order to emphasize that what we are doing here is completely classical,letusdeﬁne: ~ ~ J = ⌃ ,K= , (9.98) i 2 i i 2 i in which case ⌃i and i satisfy the algebra:

[⌃i, ⌃j ]=2i✏ijk⌃k , [⌃i, j ]=2i✏ijkk , [i, j ]=0. (9.99) which eliminates ~.Itissimpletoﬁnda4 4 matrix representation of ⌃i and i.Weﬁndtwosuch complimentary representations: ⇥

i 0 (+) 00 ( ) 0 i ⌃ = , = , = , (9.100) i 0 i 0 i 00 ✓ i◆ ✓ i ◆ ✓ ◆ both of which satisfy the set (9.99):

( ) ( ) ( ) ( ) [⌃i, j± ]=2i✏ijkk± , i± j± =0. (9.101) We also find: (+) ( ) [i , j ]=ij I + i✏ijk ⌃k . (9.102) ( ) (+) ( ) In addition, [ i ]† =i so i± is not Hermitian. Nevertheless, we can define finite transformations by exponentiation. Let us define a rotation operator U(R)by:

inˆ ⌃ ✓/2 U(R)=e · = I cos ✓/2+i(nˆ ⌃)sin✓/2 , (9.103) · ( ) and boost operators V ± (v)by:

(+) v (+)/2 (+) 10 V (v)=e · = I + v /2= , (9.104) · v/21 ✓ · ◆ and ( ) ( ) v /2 ( ) 1 v/2 V (v)=e · = I + v /2= · . (9.105) · 01 ✓ ◆ ( ) ( ) These last two equations follow from the fact that i± j± = 0. For this same reason,

( ) ( ) ( ) V ± (v0) V ± (v)=V ± (v0 + v) . (9.106)

( ) We can easily construct the inverses of V ± (v). We ﬁnd:

( ) ( ) 1 ( ) v ± /2 ( ) [ V ± (v)] = V ± ( v)=e · = I v ± . (9.107) · ( ) ( ) So the inverses of V ± (v) are not the adjoints. This means that the V ± (v) operators are not unitary. We now deﬁne combined rotation and boost operators by:

( ) ( ) ( ) 1 ( ) 1 ( ) ⇤ ± (R, v)=V ± (v) U(R) , [⇤ ± (R, v)] = U †(R)[V ± (v)] = U †(R) V ⌥ (v) . (9.108)

We ﬁnd the results:

U †(R)⌃i U(R)=Rij ⌃j , ( ) ( ) U †(R)i± U(R)=Rij j± , ( ) 1 ( ) ( ) [ V ± (v)] ⌃ V ± (v)=⌃ 2i✏ ± v , (9.109) i i ijk j k ( ) 1 ( ) ( ) ( ) [ V ± (v)] i± V ± (v)=i± , ( ) ( ) 1 U †(R) V ± (v) U(R)=V ± (R (v)) . c 2009 John F. Dawson, all rights reserved. 99 9.2. GALILEAN TRANSFORMATIONS CHAPTER 9. SYMMETRIES

So for the combined transformation,

( ) 1 ( ) ( ) [⇤ ± (R, v)] ⌃ ⇤ ± (R, v)=R( ⌃ ) 2iR( ± ) v , ⇥ (9.110) ( ) 1 ( ) ( ) ( ) [⇤ ± (R, v)] ± ⇤ ± (R, v)=R( ± ) .

( ) Comparing (9.110) with the transformations of J and K in Theorem 19, we see that ⇤ ± (R, v) are adjoint representations of the subgroup rotations and boosts, although not unitary ones. The replacement v i v ( ) ( ) ! is a reﬂection of the fact that V ± (v) is not unitary. The ⇤ ± (R, v) matrices are faithful representations of the (R, v) subgroup of the Galilean group:

( ) ( ) ( ) ( ) ⇤ ± (R0, v0)⇤ ± (R, v)=V ± (v0) U(R0) V ± (v) U(R) ( ) ( ) = V ± (v0) U(R0) V ± (v) U †(R0) U(R0) U(R) (9.111) ( ) ( ) ( ) = V ± (v0) V ± (R0(v)) U(R0R)=V ± (v0 + R0(v)) U(R0R) ( ) =⇤± (R0R, v0 + R0(v)) .

We can, in fact, display an explicit Galilean transformation for the subgroup consisting of the (R, v)elements. ( ) Let us deﬁne two 4 4 matrices X ± (x,t)by: ⇥ Deﬁnition 23. t 0 t x X(+)(x,t)= ,X( )(x,t)= . (9.112) x t 0 ·t ✓ · ◆ ✓ ◆ Then we can prove the following theorem:

( ) Theorem 23. The matrices X ± (x,t) transform under the subgroup of rotations and boosts according to:

( ) ( ) ( ) 1 ( ) ⇤ ± (R, v) X ± (x,t)[⇤ ± (R, v)] = X ± (x0,t0) , (9.113) where x0 = R(x)+vt and t0 = t.

Proof. This remarkable result is an alternative way of writing Galilean transformations for the subgroup of rotations and boosts in terms of transformations of 4 4 matrices in the “adjoint” representation. With the above deﬁnitions, the proof is straightforward and is⇥ left for the reader.

Exercise 9. Prove Theorem 23.

In this section, we have found two 4 4 dimensional matrix representations of the Galilean group. These representations turned out not to be unitary.⇥ Finite dimensional representations of the Lorentz group in relativistic theories are also not unitary. Nevertheless, ﬁnite representations of the Galilean group will be useful when discussing wave equations.

9.2.8 The massless case When M = 0, the phase for unitary representations of the Galilean group vanish, and the representation becomes a faithful one, which is simpler. For this case, the generators transform according to the equations:

U †(G) J U(G)=R J + K v¯ + a¯ P , { ⇥ ⇥ } U †(G) K U(G)=R K P ⌧ , { } (9.114) U †(G) P U(G)=R P , { } U †(G) HU(G)=H + v¯ P . · c 2009 John F. Dawson, all rights reserved. 100 CHAPTER 9. SYMMETRIES 9.3. TIME TRANSLATIONS

1 1 where v¯ = R (v) and a¯ = R (a). The generators obey the algebra:

[Ji,Jj]=i~ ✏ijkJk , [Ki,Kj]=0, [Ji,H]=0, [Ji,Kj]=i~ ✏ijkKk , [Pi,Pj]=0, [Pi,H]=0, (9.115) [Ji,Pj]=i~ ✏ijkPk , [Ki,Pj]=0, [Ki,H]=i~ Pi .

We first note that P simply rotates like a vector under the full group, so P 2 is the first Casimir invariant. We also note that if we define W = K P,then ⇥

U †(G) W U(G)=R K P ⌧ R P = R K P = R W . (9.116) { }⇥ { } { ⇥ } { } So W is a second vector which simply rotates like a vector under the full group, so W 2 is also an invariant. We also note that W is perpendicular to both P and K: W P = W K = 0. Note that W does not satisfy angular momentum commutator relations. · ·

9.3 Time translations

We have only constructed the unitary operator U(1 + G) for infinitesimal Galilean transformations. Since the generators do not commute, we cannot construct the unitary operator U(G) for a finite Galilean transformation by application of a series of infinitesimal ones. However we can easily construct the unitary operator U(G) for restricted Galilean transformations, like time, space, and boost transformations alone. We do this in the next two sections. The unitary operator for pure time translations is given by:

i H⌧ N iH ⌧/~ UH (⌧)= lim 1+ = e . (9.117) N N !1  ~ It time-translates the operator X(t) by an amount ⌧:

UH† (⌧) X(t0) UH (⌧)=X(t) ,t0 = t + ⌧, (9.118) and leaves P unchanged:

UH† (⌧) P UH (⌧)=P . (9.119) Invariance of the laws of nature under time translation is a statement of the fact that an experiment with particles done today will give the same results as an experiment done yesterday — there is no way of measuring absolute time. We ﬁrst consider transformations to a frame where we have set the clocks to zero. That is, we put t0 =0 so that ⌧ = t.Then(9.118) becomes:

X(t)=UH (t) X UH† (t)=X + V t. (9.120) where X = K/M and V = P/M . From Eq. (9.120), we ﬁnd:

X(t) U (t) x = U (t) X x = x U (t) x . (9.121) { H | i} H | i { H | i} So if we deﬁne the ket x,t by: | i x,t = U (t) x = eiHt/~ x , (9.122) | i H | i | i then (9.121) becomes an eigenvalue equation for the operator X(t) at time t:

3 X(t) x,t = x x,t , x R . (9.123) | i | i 2 Note that the eigenvalue x of this equation is not a function of t. It is just a real vector. c 2009 John F. Dawson, all rights reserved. 101 9.4. SPACE TRANSLATIONS AND BOOSTS CHAPTER 9. SYMMETRIES

From Eq. (9.122), we see that the base vector x,t satisﬁes a ﬁrst order di↵erential equation: | i d i~ x,t = H x,t , (9.124) dt | i | i and from (9.120), we obtain Heisenberg’s di↵erential equation of motion for X(t): d X(t)=[X(t),H]/i~ = P/M . (9.125) dt The general transformation of the base vectors x,t between two frames, which di↵er by clock time ⌧ only, is given by: | i x,t0 = U (t0) x = U (t0) U † (t) x,t = U (⌧) x,t , (9.126) | i H | i H H | i H | i where ⌧ = t0 t. The inner product of x,t with an arbitrary vector is given by: | i | i

(x,t)= x,t = x U † (t) = x (t) , (9.127) h | i h | H | i h | i where the time-dependent “state vector” (t) is defined by: | i iHt/ (t) = U † (t) = e ~ . (9.128) | i H | i | i This state vector satisfies a di↵erential equation given by: d i~ (t) = H (t) , (9.129) dt| i | i which is called Schrödinger’s equation. This equation gives the trajectory of the state vector in Hilbert space. Thus, we can consider two pictures: base vectors moving (the Heisenberg picture) or state vector moving (the Schrödinger picture). They are di↵erent views of the same physics. From our point of view, and remarkably, Schrödinger’s equation is a result of requiring Galilean symmetry, and is not a fundamental postulate of the theory. The state vector in the primed frame is related to that in the unprimed frame by:

(t0) = U † (t0) = U † (t0)U (t) (t) = U † (⌧) (t) , (9.130) | i H | i H H | i H | i We next turn to space translations and boosts.

9.4 Space translations and boosts

The unitary operators for pure space translations and pure boosts are built up of inﬁnitesimal transformations along any path:

N i P a iP a/ U (a)= lim 1 · = e · ~ , (9.131) P N N !1  ~ N i K v iK v/ iMv X/ U (v)= lim 1+ · = e · ~ = e · ~ , (9.132) X N N !1  ~

The space translation operator UP(a) is diagonal in momentum eigenvectors, and the boost operator UX(v) is diagonal in position eigenvectors. From the transformation rules, we have:

UP† (a) X UP(a)=X + a , (9.133)

Thus UP(a) translates the position operator and UX(v) translates the momentum operator. For the eigenvectors, this means that, for the case of no degeneracies,

x0 = x + a = U (a) x , (9.135) | i | i P | i p0 = p + Mv = U (v) p , (9.136) | i | i X | i In this section, we omit the explicit reference to w. We can ﬁnd any ket from “standard” kets x and p | 0 i | 0 i by translation and boost operators, as we did for time translations. Thus in Eq. (9.135), we set x = x0 0, and then put a x, and in Eq. (9.136), we set p = p 0, and put v p/M . This gives the relations:⌘ ! 0 ⌘ ! x = U (x) x , (9.137) | i P | 0 i p = U (p/M ) p . (9.138) | i X | 0 i We can use (9.137) or (9.138) to ﬁnd a relation between the x and p representations. We have: | i | i ip x/ x p = x U (p/M ) p = x U † (x) p = Ne · ~ , h | i h | X | 0 i h 0 | P | i where N = x0 p = x p0 . In this book,h | wei normalizeh | i these states according to the rule:

d3x, (9.139) ! x X Z d3p , (9.140) ! (2⇡ )3 p ~ X Z Then we have the normalizations:

x x0 = x p p x0 = (x x0) , (9.141) h | i h | ih | i p X 3 p p0 = p x x p0 =(2⇡~) (p p0) . (9.142) h | i h | ih | i x X This means that we should take the normalization N = 1, so that the Fourier transform pair is given by:

3 d p ip x/ (x)= x = x p p = e · ~ ˜(p) , (9.143) h | i h | ih | i (2⇡ )3 p ~ X Z 3 ip x/ ˜(p)= p = p x x = d xe · ~ (x) (9.144) h | i h | ih | i x X Z For pure space translations, x0 = x + a, wave functions in coordinate space transform according to the rule: 0(x0)= x0 0 = x U † (a)U (a) = x = (x) . (9.145) h | i h | P P | i h | i For inﬁnitesimal displacements, x0 = x +a, we have, using Taylor’s expansion, i (x +a)= x U † (a) =1+ a x P + h | P | i ~ ·h | | i ··· = 1+a + (x) . · rx ··· So the coordinate representation of the momentum operator is:

In a similar way, for pure boosts, p0 = p + Mv, wave functions in momentum space transforms according to: ˜0(p0)= p0 0 = p U † (v)U (v) = p = ˜(p) , (9.147) h | i h | X X | i h | i and we ﬁnd: ~ p X = ˜(x) . (9.148) h | | i i rp For the combined unitary operator for space translations and boosts, we note that the combined transformations give: (1, v, 0, 0)(1, 0, a, 0) = (1, v, a, 0). So, using Bargmann’s theorem, Eq. (9.26), for the phase, and Eq. (B.16) in Appendix ??,weﬁndtheresults:

1 i(Mv X P a)/~ +i 2 Mv a/~ UX,P(v, a)=e · · = e · UP(a) UX(v) , (9.149) 1 i 2 Mv a/~ = e · UX(v) UP(a) ,

So for combined space translations and boosts we ﬁnd:

Writing x0 = x + a and p0 = p + Mv, and inverting these expressions, we ﬁnd

i(Mv x+ 1 Mv a)/ x0 = e · 2 · ~ U (v, a) x , (9.150) | i X,P | i +i(p a+ 1 Mv a)/ p0 = e · 2 · ~ U (v, a) p . (9.151) | i X,P | i For combined transformations, wave functions in coordinate and momentum space transform according to the rule:

+i(Mv x+ 1 Mv a)/ 0(x0)= x0 0 = x0 U (v, a) = e · 2 · ~ (x) , (9.152) h | i h | X,P | i i(p a+ 1 Mv a)/ ˜0(p0)= p0 0 = p0 U (v, a) = e · 2 · ~ ˜(p) . (9.153) h | i h | X,P | i These functions transform like scalars, but with an essential coordinate or momenutm dependent phase, characteristic of Gailiean transformations.

Example 29. It is easy to show that Eq. (9.152), is the Fourier transform of (9.153),

3 d p0 ip0 x0/ 0(x0)= e · ~ ˜0(p0) (2⇡ )3 Z ~ 3 +i(Mv x+ 1 Mv a)/ d p ip x/ +i(Mv x+ 1 Mv a)/ = e · 2 · ~ e · ~ ˜(p)=e · 2 · ~ (x) . (2⇡ )3 Z ~ as required by Eq. (9.143).

We discuss the case of combined space and time translations with boosts, but without rotations, in Appendix ??. We turn next to rotations. c 2009 John F. Dawson, all rights reserved. 104 CHAPTER 9. SYMMETRIES 9.5. ROTATIONS

9.5 Rotations

In this section, we discuss pure rotations. Because of the importance of rotations and angular momentum in quantum mechanics, this topic is discussed in great detail in Chapter ??. We will therefore restrict our discussion here to general properties of pure rotations and angular momentum algebra.

9.5.1 The rotation operator

The total angular momentum is the sum of orbital plus spin: J = L + S,with[Li,Sj ] = 0. Common eigenvectors of these two operators are then the direct product of these two states:

`, m ; s, m = `, m s, m . (9.154) | ` s i | ` i| s i The rotation operator is given by the combined rotation of orbital and spin operators:

inˆ J ✓/~ inˆ L ✓/~ inˆ S ✓/~ UJ(R)=e · = e · e · = UL(R) US(R) . (9.155)

The orbital rotation operator acts only on eigenstates of the position operator X, or momentum operator P, For pure rotations, the rotation operator can be found by N sequential inﬁnitesimal transformations ✓ = ✓/N about a ﬁxed axis nˆ:

i n J ✓ N inˆ J ✓/~ UJ(nˆ,✓)= lim 1+ · = e · . (9.156) N N !1  ~ For pure rotations, the Galilean phase factor is zero so that we have:

UJ(R0) UJ(R)=UJ(R0R) . (9.157)

From Theorem 19 and Eq. (9.59), for pure rotations, we have:

U †(nˆ,✓) J U (nˆ,✓)=R (nˆ,✓) J J (nˆ,✓) . (9.158) J i J ij j ⌘ i th We discuss parameterizations of the rotation matrices R(nˆ,✓) in Appendix ??. Here Ji(nˆ,✓)isthei component of the operator J evaluated in the rotated system. Setting i = z, we ﬁnd for the z-component:

Jz(nˆ,✓) UJ†(nˆ,✓)=UJ†(nˆ,✓) Jz (9.159)

2 2 2 2 We also know that J = Jx + Jy + Jz is an invariant:

2 2 UJ†(nˆ,✓) J UJ(nˆ,✓)=J . (9.160)

So from Eq. (9.159), we ﬁnd that:

Jz(nˆ,✓) U †(nˆ,✓) j, m = ~ m U †(nˆ,✓) j, m , (9.161) J | i J | i from which we conclude that the quantity in brackets is an eigenvector of Jz(nˆ,✓) with eigenvalue ~m. That is, we can write: j, m(nˆ,✓) = U †(nˆ,✓) j, m . (9.162) | i J | i It is also an eigenvector of J 2 with eigenvalue 2 j(j + 1). It is useful to deﬁne a rotation matrix D(j) (nˆ,✓) ~ m0,m by: (j) D (nˆ,✓)= jm U (nˆ,✓) jm0 . (9.163) m,m0 h | J | i c 2009 John F. Dawson, all rights reserved. 105 9.5. ROTATIONS CHAPTER 9. SYMMETRIES

Matrix elements of the rotation operator are diagonal in j. The rotation matrices have the properties:

j (j) (j) D (R) D ⇤ (R)= , (9.164) m,m0 m00,m0 m,m00 m = j X0 (j) (j) 1 m0 m (j) Dm,m⇤ (R)=Dm ,m(R )=( ) D m, m (R) . (9.165) 0 0 0 We can express nˆ,✓; j, m in terms of the rotation matrices. We write: | i j (j) j, m(nˆ,✓) = D ⇤ (nˆ,✓) j, m0 . (9.166) | i m,m0 | i m = j X0

In the coordinate representation of orbital angular momenta, spherical harmonics are deﬁned by: Y`,m(⌦) = ⌦ `, m . Using Eq. (9.166), we ﬁnd: h | i

Y (⌦0)= ⌦0 `, m = ⌦ U †(nˆ,✓) `, m `,m h | i h | J | i ` ` (`) (`) (9.167) = D ⇤ (nˆ,✓) ⌦ `, m0 = D ⇤ (nˆ,✓) Y`,m (⌦) , m,m0 h | i m,m0 0 m = ` m = ` X0 X0 where ⌦and ⌦0 are spherical angles of the same point measured in two di↵erent coordinate systems, rotated relative to each other.

9.5.2 Rotations of the basis sets Now L and therefore J does not commute with either X or P. Therefore they cannot have common eigenvectors. However S does commute with with both X or P. Supressing the dependence on w and M, the common eigenvectors are: x,sm , and p,sm . (9.168) | i | i A general rotation of the ket x,sm can be obtained by ﬁrst translating to the state where x = 0, then rotat- | i ing, and then translating back to a rotated state x0 = R(x). That is, (R, 0, 0, 0) = (1, x0, 0, 0)(R, 0, 0, 0)(1, x, 0, 0). The trick is that the orbital angular momentum operator L acting on a state with x = 0 gives zero, so on this state J = S. The phases all work out to be zero in this case, so we ﬁnd:

(s) U †(R) x0,sm0 = D ⇤ (R) x,sm , (9.170) J | i m0,m | i m X which gives: (s) 0 (x0)= D (R) (x) , (9.171) sm0 m0,m sm m X where x,sm = (x)with 0 = U(R) . h | i sm | i | i c 2009 John F. Dawson, all rights reserved. 106 CHAPTER 9. SYMMETRIES 9.6. GENERAL GALILEAN TRANSFORMATIONS

9.6 General Galilean transformations

The general Galilean transformation for space and time translations and rotations is given by:

x0 = R(x)+vt + a ,

t0 = t + ⌧. (9.172)

Starting from the state sm; x,t , we generate a full Galilean transformation G =(R, v, a,⌧) by ﬁrst doing a time translation back| to t = 0,i a space translation back to the origin x = 0, then a rotation (which now can be done with the spin operator alone), then a space translation to the new value x0, then a boost to the v frame, and ﬁnally a time translation forward to t0. This is given by the set:

G =(1, 0, 0,t0)(1, v, 0, 0)(1, 0, x0, 0)(R, 0, 0, 0)(1, 0, x, 0)(1, 0, 0, t) , =(1, 0, 0,t0)(1, v, 0, 0)(1, 0, x0, 0)(R, 0, 0, 0)(1, 0, x, t) , =(1, 0, 0,t0)(1, v, 0, 0)(1, 0, x0, 0)(R, 0, R(x), t) , =(1, 0, 0,t0)(1, v, 0, 0)(R, 0, x0 R(x), t) , =(1, 0, 0,t0)(R, v, x0 R(x) vt, t) , =(R, v, a,⌧) , (9.173) as required. The combined unitary transformation for the full Galilean group is then given by:

ig(x,t)/ U (t0) U (v)U (x0) U (R) U ( x) U ( t)=e ~ U(G) . (9.174) H X P J P H The only contribution to the phase comes from between step four and step ﬁve in the above. Using Bargmann’s theorem, we ﬁnd:

1 1 2 1 g(x,t)= M v (x0 R(x)) = Mv t + M v a . (9.175) 2 · 2 2 · So

ig(x,t)/ U(G) x,t; sm = e ~ U (t0) U (v)U (x0) U (R) U ( x) U ( t) x,t; sm | i H X P J P H | i ig(x,t)/ = e ~ U (t0) U (v)U (x0) U (R) U ( x) x, 0; sm H X P J P | i ig(x,t)/ = e ~ U (t0) U (v)U (x0) U (R) 0, 0; sm H X P J | i ig(x,t)/ = e ~ U (t0) U (v)U (x0) U (R) 0, 0; sm H X P S | i ig(x,t)/~ (s) = e UH (t0) UX(v)UP(x0) 0, 0; sm0 D (R) | i m0,m Xm0 ig(x,t)/~ (s) = e UH (t0) UX(v) x0, 0; sm0 D (R) | i m0,m Xm0 ig(x,t)/~ iMv x0 (s) = e UH (t0) e · x0, 0; sm0 D (R) | i m0,m Xm0 if(x,t)/ (s) = e ~ x0,t0; sm0 D (R) (9.176) | i m0,m Xm0 Where we have deﬁned the phase factor (G)by:

Inverting Eq. (9.176), we ﬁnd:

if(x,t)/ (s) U †(G) x0,t0; sm0 = e ~ x,t; sm D ⇤ (R) . (9.178) | i | i m0,m m X So that: if(x,t)/ (s) 0 (x0,t0)=e ~ D (R) (x,t) . (9.179) sm0 m0,m sm m X where sm(x,t)= x,t; sm , and we have put: 0 = U(G) . It is important to note here that the phase factor f(x,t)h depends| oni x and t, as well as the| parametersi | ofi the Galilean transformation.

Exercise 10. Find the general Galilean transformation of momentum eigenvectors: p,sm . Show that the | i transformed functions ˜sm(p) give the same result as as the Fourier transform of Eq. (9.179).

9.7 Improper transformations

In this section we follow Weinberg[?, p. 77]. We ﬁrst extend the kinds of Galilean transformations we consider to include parity, time reversal, and charge conjugation. The full Galilean transformations are now described by: x0 = rR(x)+vt + a ,t0 = t + ⌧. (9.180) Here r =det[R ] and  can have values of 1. We still require that lengths are preserved so that R is still orthogonal, and that the rate of passage± of time does not dilate or shrink, only the direction of time can be reversed. So the full group, including improper transformations, is now represented by the twelve parameters: G =(R, v, a,⌧,r,) . (9.181) The full group properties are now stated in the next theorem.

Theorem 24. The composition rule for the full Galilean group is given by:

G00 = G0G =(R0, v0, a0,⌧0,r0,0 )(R, v, a,⌧,r,) (9.182) =(R0R,v0 + r0R0(v), a0 + v0⌧ + r0R(a),⌧0 + ⌧,r0r, 0, )

Proof. The proof follows directly from the complete transformation equations (9.180) and left as an exercise.

9.7.1 Parity In this section we consider parity transformations (space reversals) of the coordinate system. This is represented by the group elements: G =(1, 0, 0, 0, 1, +1) . (9.183) P 1 We note that GP = GP . So using the rules given in Theorem 24, we ﬁnd for the combined transformation:

1 G0 = G GGP =(1, 0, 0, 0, 1, +1) ( R, v, a,⌧,r,)(1, 0, 0, 0, 1, +1) P (9.184) =(R, v, a,⌧,r,) . The phase factors are zero in this case. So we have:

Now if we take r = 1 and  = 1, both G and G0 are proper. This means that we can take G =1+G, where G =(!,v, a, ⌧,1, 1). Then G0 =1+G0,whereG0 =(!, v, a, ⌧,1, 1). So then U(1 + G) can be represented by:5 i U(1 + G)=1+ ✓ nˆ J +v K a P +⌧H + . (9.186) ~ · · · ··· n o Using this in Eq. (9.185), we ﬁnd: 1 J = J , P P 1 K = K , P P 1 (9.187) P = P , P P 1 H = H. P P 1 We note that is linear and unitary, with eigenvalues of unit magnitude. We also have: = † = . We assume thatP the Casimir invariants M and W remain unchanged by a parity transformation.P P P Exercise 11. Show that under parity, 1 X(t) = X(t) , (9.188) P P where X(t)=X + V t,whereX = K/M and V = P/M . We discuss the action of parity on eigenvectors of angular momentum in Section 21.1.4.

9.7.2 Time reversal Time reversal is represented by the group elements: G =(1, 0, 0, 0, +1, 1) , (9.189) T 1 with GT = GT . So again using the rules given in Theorem 24, we find for the combined transformation: 1 G0 = G GGT =(1, 0, 0, 0, +1, 1) ( R, v, a,⌧,r,)(1, 0, 0, 0, +1, 1) T (9.190) =(R, v, a, ⌧,r,) . So we have: 1 1 U(G) = U(G GG )=U(G0) . (9.191) T T T T Again, we take r = +1 and  = +1, so that G =1+G and G0 =1+G0,where G = !,v, a, ⌧,1, 1 , (9.192) G0 = !, v, a, ⌧,1, 1 , Both of these transformations are proper. So we can take U(G) and G(G0)toberepresentedbytheinfinites- 1 imal form of Eq. (9.186). Since we will require to be anti-linear and anti-unitary, i = i, and, using (9.191), we find: T T T 1 J = J , T T 1 K = K , T T 1 (9.193) P = P , T T 1 H = H. T T We also assume that M and W are unchanged by a time-reversal transformation. The eigenvalues of are 1 T also of unit magnitude. We also have: = † = . We discuss time reversal of angular momentum eigenvectors in Section 21.1.4. T T T 5We do not use the extended group in this discussion. c 2009 John F. Dawson, all rights reserved. 109 9.7. IMPROPER TRANSFORMATIONS CHAPTER 9. SYMMETRIES

Exercise 12. Show that under time reversal,

1 X(t) = X( t) , (9.194) T T where X(t)=X + V t,whereX = K/M and V = P/M . For combined parity and time-reversal transformations, we ﬁnd:

1 ( ) J ( )= J , PT PT 1 ( ) K ( )= K , PT PT 1 (9.195) ( ) P ( )=P , PT PT 1 ( ) H ( )=H. PT PT 9.7.3 Charge conjugation The charge conjugation operator changes particles into antiparticles. This is not a space-time symmetry, but one that reverses the sign of theC mass and spin. That is, we assume that:

1 1 M = M, S = S . (9.196) C C C C In addition, we take to be linear and unitary, and: C 1 J = J , C C 1 K = K , C C 1 (9.197) P = P , C C 1 H = H. C C The eigenvalues of are again of unit magnitude. If we deﬁne X = K/M , and V = P/M , then this means that C 1 X = X , C C 1 (9.198) V = V , C C So we have the following theorem: Theorem 25 ( ). From Eqs. (9.195) and (9.197), the combined ( ) operation when acting on the generators of thePT Galilean C transformation, leaves the generators unchanged:PT C

1 ( ) J ( )=J , PT C PT C 1 ( ) K ( )=K , PT C PT C 1 (9.199) ( ) P ( )=P , PT C PT C 1 ( ) H ( )=H. PT C PT C That is, the generators are invariant under ( ). PT C Exercise 13. Show that under charge conjugation,

1 X(t) = X( t) , (9.200) C C where X(t)=X + V t,withX = K/M and V = P/M . So when acting on the equation of motion of X(t), charge conjugation has the same e↵ect as time reversal. We can interpret this as meaning that in non-relativistic physics, we can think of an antiparticle as a negative mass particle moving backwards in time. c 2009 John F. Dawson, all rights reserved. 110 CHAPTER 9. SYMMETRIES 9.8. SCALE AND CONFORMAL TRANSFORMATIONS

Let us be precise. If represents a single particle state, then c = is the charge conjugate state. Ignoring spin for the moment,| i if m ,w ,E ; x,t are eigenstates| of iX(tC|) andi M with positive eigenvalues | 0 0 0 i m = m0 > 0, w = w0 > 0 and E = E0 > 0, then m ,w ,E ; x,t = m , w , E ; x,t , (9.201) C| 0 0 0 i | 0 0 0 i is an eigenvector X(t), M, W , and H with negative eigenvalues m = m0 < 0, w = m0 < 0, and E = E < 0. So the charge conjugate wave function with m, w, and E all positive: 0 (m ,w ,E ; x,t)= m ,w ,E ; x,t = m ,w ,E ; x,t c 0 0 0 h 0 0 0 | c i h 0 0 0 |C| i = m , w , E ; x,t = ( m , w , E ; x,t) , (9.202) h 0 0 0 | i 0 0 0 is the same as the wave function with m0, w0, and E0 negative. We will study single particle wave functions in the next chapter. Charge conjugate symmetry says that, in priciple, we cannot tell the di↵erence between a world consisting of particles or a world consisting of antiparticles.

9.8 Scale and conformal transformations

Scale transformations are changes in the measures of length and time. An interesting question is if there are ways to determine a length or time scale in absolute terms, or are these just arbitrary measures. If there are no physical systems that can set these scales, we say that the fundamental forces in Nature must be scale invariant. Conformal invariance is a combined space-time expansion of the measures of length and time, and generalizes scale changes. We discuss these additional space-time symmetries in the next two sections.

9.8.1 Scale transformations Scale transformations are of the form:

xi0 = ↵xi ,t0 = t. (9.203) We require, in particular, that if (x,t) satisfies Schrödinger’s equation with w = 0 for a spinless free particle in ⌃, then 0(x0,t0) satisfies Schrödinger’s equation in ⌃0. Probability must remain the same, so we require that 2 3 2 3 0(x0,t0) d x0 = (x,t) d x. (9.204) | | | | With this observation, it is easy to prove the following theorem.

Theorem 26. Under scale transformations x0 = ↵x and t0 = t, spinless scalar solutions of Schr¨odinger’s equation transform according to:

3/2 ig(x,t)/ 0(x0,t0)=↵ e ~ (x,t) . (9.205) with = ↵2 and g(x,t)=C, a constant phase. Exercise 14. Prove Theorem 26. We put ↵ = es and then = e2s, so that infinitesimal scale transformations become: x =s x , t =2st. (9.206) We now follow our work in example 27 to find a di↵erential representation of the scale generator D. Using Eq. (9.205), infinitesimal scale changes of scalar functions are given by:

3s/2 0(x0,t0)=e (x0 x,t0 t) = 1 3s/2+ 1 s x s 2 t@ + (x0,t0) ··· · r t ··· (9.207) = 1 s 3/2+x +2t@ + (x0,t0) · r t ··· n o c 2009 John F. Dawson, all rights reserved. 111 9.8. SCALE AND CONFORMAL TRANSFORMATIONS CHAPTER 9. SYMMETRIES

The dilation generator D is deﬁned by:

(x,t)= 0(x,t) (x,t)= i sD (x,t) , (9.208) from which we ﬁnd: 3 1 3 D = i + x 2it@ = i + x P 2 tH . (9.209) 2 i · r t 2 · We can drop the factor of 3i/2 since this produces only a constant phase. Using the di↵erential representations in Eqs. (9.13), we ﬁnd the commutation relations for D:

[ D, P ]=iP , [ D, H ]=2iH , [ D, K ]= iK , (9.210) i i i i and commutes with Ji. D also commutes with M, but we note that the ﬁrst Casimir operator W = H = P 2/2M does not commute with D. In fact, we ﬁnd:

[ D, W ]=2iW . (9.211)

So the internal energy W breaks scale symmetry.

9.8.2 Conformal transformations Conformal transformations are of the form:

xi t x0 = ,t0 = , (9.212) i 1 ct 1 ct where c has units of reciprocal time (not velocity!) and can be positive or negative. Note that 1/t0 =1/t c. For a scalar spin zero free particle satisfying Schr¨odinger’s equation, probability is again conserved according to (9.204), and we ﬁnd the following result for conformal transformations:

Theorem 27. Under scale transformations x0 = ↵x and t0 = t, spinless scalar solutions of Schr¨odinger’s equation transform according to:

3/2 ig(x,t)/ 0(x0,t0)=(1 ct) e ~ (x,t) . (9.213) where 1 mc x2 g(x,t)= . (9.214) 2 1 ct Exercise 15. Prove Theorem 27. For this, it is useful to note that:

2 0 =(1 ct) ,@0 =(1 ct) @ c(1 ct) x . (9.215) r r t t · r and that: ~ ~ eig(x,t)/~ (x,t) = eig(x,t)/~ +( g(x,t)) (x,t) . (9.216) i r i r r h i h i Inﬁnitesimal conformal transformations are given by:

x =ctx , t =ct2 . (9.217)

So from Eq. (9.213), inﬁnitesimal conformal transformations of scalar functions are given by:

3 ~ 2 0(x0,t0)= 1 t c + 1+ mx c + 2 ··· 2i ··· n on 2 o 1 ctx ct @ + (x0,t0) (9.220) ⇥ · r t ··· n 3 ~ 2 o 2 = 1+c t + mx tx t @ + (x0,t0) , 2 2i · r t ··· n n o o The conformal generator C is deﬁned by:

(x,t)= 0(x,t) (x,t)=i cC (x,t) , (9.221) from which we ﬁnd: 3i ~ t C = t mx2 + x it2 @ 2 2 i · r t 3i ~ = t mx2 + t x P t2 H (9.222) 2 2 · 3i ~ = t mx2 + tD+ t2 H. 2 2 We ﬁnd the following commutation relations for C:

[ C, H ]= iD, [ C, D ]= 2iC, (9.223) and commutes with all other operators. Note that scale and conformal transformations do not commute. So if we put: 1 1 1 G = (H + C) ,G= (H C) ,G= D, (9.224) 1 2 2 2 3 2 we ﬁnd that G satisﬁes a O(2, 1) algebra:

[ G ,G ]= iG , [ G ,G ]=iG , [ G ,G ]=iG . (9.225) 1 2 3 1 3 2 2 3 1 Since [ G ,J ] = 0, the group structure of the extended group has O(3) O(2, 1) symmetry. i j ⇥ 9.9 The Schr¨odinger group

The extension of the Galilean group to include scale and conformal transformations is called the Schr¨odinger or non-relativistic conformal group, which we write as . We consider combined scale and conformal transformations of the following form: S R(x)+vt + a ↵t + x0 = ,t0 = ,↵ =1. (9.226) t + t + Here ↵, , , and are real parameters, only three of which are independent. This transformation contains both scale and conformal transformations as special interrelated cases. The group elements now consist of twelve independent parameters, but it is useful to write them in terms of thirteen parameters with one constraint: S =(R, v, a,↵,,,). The extended transformation is a group. The group multiplication properties are contained in the next theorem: Theorem 28. The multiplication law for the Schr¨odinger group is given by:

S00 = S0S =(R0, v0, a0,↵0,0,0,0)(R, v, a,↵,,,)

=(R0R, R0(v)+↵v0 + a0,R0(a)+v0 + a0, (9.227)

A faithful ﬁve-dimensional matrix representation is given by:

R va S = 0 ↵ ,S00 = S0S , (9.228) 0 0 1 @ A which preserves the determinant relation: det[ S ]=↵ = 1. The unit element is 1 = (1, 0, 0, 1, 0, 0, 1) and the inverse element is:

1 1 1 1 1 1 S =(R , R (v)+R (a), ↵R (a)+R (v),, , ,↵) . (9.229) For inﬁnitesimal transformations, it is useful to write:

↵ =1+s + , ··· =⌧ + , ··· (9.230) = c + , ··· =1 s + , ··· so that

↵ =(1+s + )(1 s + ) (⌧ + )( c + )=1+O(2) , (9.231) ··· ··· ··· ··· as required. ⌧,s, and c are now independent variations. So the unitary transformation transformation for inﬁnitesimal transformations is now written as: i U(1 + S)=1+ ✓ nˆ J +v K a P +⌧H+sD cC + , (9.232) ~ · · · ··· n o in terms of the twelve generators J, K, P, H, D, and C.

References

[1] E. P. Wigner, Gruppentheorie und ihre Anwendung auf dei Quantenmechanic der Atomspektren (Braun- schweig, Berlin, 1931). English translation: Academic Press, Inc, New York, 1959. [2] V. Bargmann, “On unitary ray representations of continuous groups,” Ann. Math. 59, 1 (1954). [3] J.-M. Levy-Leblond, “Galilei group and nonrelativistic quantum mechanics,” J. Math. Phys. 4, 776 (1963).

[4] J.-M. Levy-Leblond, “Galilean quantum ﬁeld theories and a ghostless Lee model,” Commun. Math. Phys. 4, 157 (1967). [5] J.-M. Levy-Leblond, “Nonrelativistic particles and wave equations,” Commun. Math. Phys. 6, 286 (1967).

[6] J.-M. Levy-Leblond, “Galilei group and galilean invariance,” in E. M. Loebl (editor), “Group theory and its applications,” volume II, pages 222–296 (Academic Press, New York, NY, 1971).

Citation: J. Math. Phys. 4, 776 (1963); doi: 10.1063/1.1724319 View online: http://dx.doi.org/10.1063/1.1724319 View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v4/i6 Published by the American Institute of Physics.

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Downloaded 13 Nov 2012 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions JOURNAL OF MATHEMATICAL PHYSICS VOLUME 4, NUMBER 6 JUNE 1963

Galilei Group and Nonrelativistic Quantum Mechanics

JEAN-MARC LEVy-LEBLOND Laboratoire de Physique Theorique et Hautes Energies, Orsay, France (Received 16 January 1963)

This paper is devoted to the study of the GaJilei group and its representations, The Galilei group presents a certain number of essential differences with respect to the Poincare group, As Bargmann showed, its physical representations, here explicitly constructed, are not true representations but only up-to-a-factor ones, Consequently, in nonrelativistic quantum mechanics the mass has a very special role, and in fact, gives rise to a superselection rule which prevents the' existence of unstable particles. The internal energy of a nonrelativistic system is known to be an arbitrary parameter' this is shown to come also from Galilean invariance, because of a nontrivial concept of equivalence bet~een physical representations. On the contrary, the behavior of an elementary system with respect to rotations, is very similar to the relativistic case. We show here, in particular how the number of polarization states reduces to two for the zero-mass case (though in fact there 'are no physical zero mass systems in nonrelativistic mechanics). Finally, we study the two-particle system where the orbital angular momenta quite naturally introduce themselves through the decomposition of the tensor product of two physical representations.

INTRODUCTION Inonu and Wigner3 have indeed shown that under INCE the work by Wignerl came out, the no condition can the basis functions of the Galilei SPoincare group (inhomogeneous Lorentz group) group true representations be interpreted as wave and its unitary representations have become well functions of physical particles. With these functions, known. In particular, each relativistic wavefunction one can construct neither localized states, nor even corresponds to some unitary representation of the states with definite velocity. Conversely, Hamer 4 Poincare group, and in a certain sense, one usually mesh studying the infinitesimal group operations, says that an elementary particle is associated to a has shown that one can construct a position operator unitary irreducible representation of the group. only in the case of nontrivial projective representa Within such a definition, an elementary particle tions. One easily sees that the solutions of the is characterized by its mass and spin. Schrodinger equation for a free particle transform It was much later that such a work was under precisely according to such representations. taken for the Galilei group, the invariance group In the following pages, these unitary, irreducible, of nonrelativistic mechanics. The Galilei group has, nontrivial projective representations of the Galilei in fact, a rather more intricate structure than has group will be called, for short, "physical representa the Poincare group and this has important repercus tions." sions in the study of the group representations. In the first section, we recall some generalities Indeed, in quantum mechanics, we deal with the about the Galilei group and its structure. In the unitary projective (i.e. up-to-a-factor) representa second one, we explicitly construct the physical tions of the group concerned. But in most of the representations of the Galilei group, with the help 6 physically interesting cases, as Bargmann showed,2 of the "little group" technique. The third section the study of unitary projective representations of is devoted to a physical discussion of these rep the group can be reduced to the study of true unitary resentations where we exhibit their connection with representations of its universal covering group. Such the free-particle Schrodinger equation, and obtain is the case of the rotation, Lorentz, and Poincare a group-theoretical characterization of a non groups. relativistic elementary system by its spin, mass and On the other hand, as Bargmann also showed,2 internal energy-this last parameter in fact revealing the Galilei group owns a (one-dimensional) infinity itself to be arbitrary. We next study the zero-mass of projective representations classes, nonequivalent case. In this fourth section, we rediscover some of to true representations, and, what is more trouble the true representations already studied by Inonu some, the physically meaningful representations are 3 E. Inonu and E. P. Wigner, Nuovo Cimento 9,705 (1952). precisely these nontrivial projective representations. • M. Ham~rmesh, Ann. Phys. 9, 518 (1960). 6 See, for mstll:nc~, M. Hamermesh, Group Theory (Addi 1 E. P. Wigner, Ann. Math. 40, 149 (1939). son-Wesley PublIshmg Company, Inc., Reading, Massa 2 V. Bargmann, Ann. Math. 59, 1 (1954). chusetts, 1962), Sec. 12-7. 776

Downloaded 13 Nov 2012 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions GAL I LEI G R 0 U PAN D NON R E LA T I V 1ST Ie QUA N TUM ME C HAN I C S 777 and Wigner. a One can then give to these representa In other words, the Poincare group can be written tions some vague physical meaning, and we observe,

of rule). In the sixth and last section, we decompose a simple group ffi. the tensor product of two physical representations We will now use the following result, due to of the Galilei group into a direct sum (integral) of Bargmann,2 which we reproduce without proof: such physical (irreducible) representations. "The physical representations of the Galilei group I. THE GALILEI GROUP are obtained from the projective unitary representa The proper Galilei group to which we restrict tions of its universal covering group characterized ourselves (we exclude the inversions) contains the by the system of factors: translations in space and time, the rotations and w(G', G) = exp [i(tm) (a' ·R'v - v' ·Ra the pure Galilei transformations, i.e., transitions to a uniformly moving coordinate system. Let us note + bv' ·R'v)], (1.8) the general element of the group by where G = (b, a, v, R), G' = (b', a', v', R') and G = (b, a, v, R), (1.1) m is any real number. where b is a time translation, a a space translation, This means that to each element G of the universal v a pure Galilei transformation, and R a rotation. covering group (which one obtains merely by replac The group acts on the coordinates (x, t) of an event ing the rotations R by the elements of the unitary in space-time according to unimodular group), corresponds a unitary operator UCG) such that the multiplication law x' = Rx + vt + a, t' = t + b. (1.2) U(G') U(G) = w(G', G) U(G'G) (1.9) We thus get the multiplication law for the group: holds, where G', G, w(G', G) have been defined above. G'G = (b', a', v', R')(b, a, v, R) The ensuing transition from the covering group = (b' + b, a' + R'a + bv', v' + R'v, R'R). (1.3) to the original Galilei group only adds a possible sign ambiguity. The identity for the group is In a more elaborate language, we are looking for 1 = (0, 0, 0, 1), (1.4) true unitary representations of some nontrivial central extension of the Galilei group universal and the inverse element of G = (b, a, v, R) is given by covering group by a one-dimensional abelian group. 6 G- 1 = (-b, -R-1(a - bv), -R-1v, R-1). (1.5) Let us note One notices at once the complexity of the Galilei G=(O,G), Oreal, (1.10) group structure. which are the elements of this extension. We have The Poincare group

, the time translations, and it is only 6 F. Lurcat and L. Michel (unpublished). L. Michel, the factor group (g/'U)/5:> which is a simple group Lectures at the Istanbul Summer School (1962) (to be ffi, the rotation group. published).

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deal now, has a structure rather different from the first the following equalities, inferred from (1.3), one of the original Galilei group. In fact, the maximal (1.8), and (1.9): abelian invariant subgroup of 9 is the space-time e, U(b, a, v, R) = exp (-i(!m)a·v) translations plus the one-parameter central sub group. The factor group 9/ e in turn admits a X U(b, a, 0, l)U(O, 0, v, R), (II.2) maximal abelian invariant subgroup 'U, made up U(b, a, v, R) = exp [i(!m)(a·v - bv.v)] of the pure Galilei transformations (3 parameters). Finally, the factor group C§/e)/'U is a simple group X U(O, 0, v, R)U(b, R-1(a - by), 0,1), (II.2') m, the rotation group. We can write whence 9 = (m X 'U) X e, (1.13) U(b, a, 0, l)U(O, 0, v, R) = exp [im(a·v - !bv·v)] where the products are semidirect products. This X U(O, 0, v, R)U(b, R-1(a - by), 0,1). (II.3) is the structure we shall be concerned with. Let us finally notice that setting m = ° in (1.8) brings Letting each member of this equality between us back to the study of true representations of the operators act upon some basis function 1/;(p, E, r), Galilei group: and taking (11.1) into account, we get m = °==} w(G', G) = 1 ==} U(G') U(G) U(b, a, 0, l)U(O, 0, v, R)1/;(p, E, r = U(G'G). (1.14) = exp [im(a·v - !bv·v)] The central extension 9 becomes a trivial one X exp [-ibE + i(R-1a·bR-1v).p] (direct product). Unless otherwise specified, we will deal exclusively X U(O, 0, v, R)1/;(p, E, r), (II .4) from now on with the case m ~ 0. U(b, a, 0, l)U(O, 0, v, R)1/;(p, E, r) 2 II. PHYSICAL REPRESENTIONS OF THE = exp [-ib(E + v·Rp + !mv ) + ia·(Rp + my)] GALILEI GROUP X U(O, 0, v, R)1/;(p, E, r). (II.5) We now proceed to construct the physical (i.e., irreducible, unitary, nontrivial projective) rep That is to say, the function U(O, 0, v, R)1/;(p, E, n resentations of the Galilei group, making use of its transforms according to the representation (p', E') structure as studied above and following the "little of the space-time translations group, where group" technique.s p' = Rp + mY, Let us suppose we have found some physical (II.6) 2 representations of the Galilei group. If we restrict E' = E + v·Rp + !mv • ourselves to the abelian subgroup of space-time e Thus, if (p', E') and (p, E) are connected by the translations, this representation will decompose into relation a direct integral of unitary irreducible representa tions of the subgroup e. These representations are E' - (p'2/2m) = E _ (p' /2m) , (11.7) well-known, they are designated by a real vector p and a real number E. We can then choose as a set it is always possible to find an element (0, 0, v, R) of basis functions, square-integrable functions of the Galilei group (more precisely of the factor group g/e) such that (II.6) holds. 1/;(p, E, n, where r is an additional set of variables which may be needed to distinguish the basis In other words, if a physical representation of functions belonging to the same irreducible rep the Galilei group contains an irreducible representa resentation of the translation group e. We now tion (p, E) of the translation group, it contains all know the representation of this subgroup: the representations (p', E') given by (11.7). There fore, there is a one-to-one correspondence between U(b, a, 0, l)1/;(p, E, r) the points of the paraboloid: = exp (-ibE + ia·p)lf(p, E, t). (11.1) E - (p2/2m) = 'V = ct., (II. 7') Using the mUltiplication law of the group representa and the irreducible representations of the transla tion (1.9), a factor system, i.e., a real number m tion group contained in a physical representation (1.8) having been chosen, we look for the rep of the Galilei group. resentation of the factor group g/ e. Let us note This enables us to construct the Hilbert space X

Downloaded 13 Nov 2012 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions GAL I LEI G R 0 UP AND NON R E L A T I V 1ST I C QUA N TUM M E C HAN I C S 779 of the representation as a direct integral of the First we set up a point (Po, Eo) on the paraboloid. Hilbert spaces JCp,E where acts the irreducible rep Then for each point (p, E) of that same paraboloid, resentation (p, E) of e: one can select one element (11.11) Je = J df..LCp, E) Jep,E, (11.8) of the Galilei group such that V pE acting on (Po, Eo) where J.I.(p, E) is an invariant measure on the according to (11.6) transforms it into Cp, E), which paraboloid (II.8). As a matter of fact, we denote by (II. 11') df..LCp, E) = dp dE orE - Cp2/2m) - 'U]. (II.9) N ow let V be any element of the factor group: The Little Group V = (0, 0, v, R), (11.12) We now search for the "little group",6 i,e., the subgroup ~(p, E) of the Galilei group constituted and let (p', E') be the result of the action of V by those elements (0, 0, v, R) of the factor group (p, E) according to (II.6): s/e such that the function U(O, 0, v, R)if;(p, E, r) V(p, E) = (p', E'). (I 1. 13) still belongs to the irreducible representation (p, E) of e. ~(p, E) is what mathematicians call the This can also be written as "stabilisator" of (p, E). After (II.6), we get for VVpE(Po, Eo) = Vp'E'(PO' Eo), (0, 0, v, R) the conditions or p Rp mY, = + (II.lO) 2 E = E + V· Rp + !mv , Thus, (II.14) which can also be written as is an element of the little group ~(po, Eo), which, p = Rp mY, + (II.10') of course, depends on (p, E) and V. p2 = (Rp + mv)2. Conversely, every element V of the factor group Sl e can be written in the form These two conditions then are not independent, the second being implied by the first one. (11.15) Thus an element of the little group ~(p, E) is with the definitions (11.11'), (II.I2), and (II.I3). uniquely defined by the choice of a rotation R, Now, after (II.5), (II.6), and (II.ll'), U(Vp,B)' since in that case, the condition (IL10') determines v. if; (Po, Eo) is proportional to if;(p, E). The simplest This correspondence between the little group choice is then to define U(Vp,E) by l)(p, E) and the rotation group (R is an isomorphism. Indeed, (II.I6) Finally, we choose a representation D' of the little p = Rp + mY, p = R'p + my' group. The variable r is merely an index, running implies from -8 to +8, on which act the (28 + I)-dimen p = R'(Rp + my) + mv' = R'Rp + m(R'v + v'). sional matrices of the representation D': Vo E f)(po, Eo) :==} U(Vo)if;(po, Eo, S) The product of (0, 0, v', R') and (0, 0, v, R) cor responds to the product of Rand R'. H = L if;(Po, Eo, ~)[D'(VO)]~i' (II.17) The little group representations are then well ~--. known: they are the rotation group representations. Therefore, for any V = (0, 0, v, R) and any The irreducible ones are labeled by an integer of if;(p, E, r), we have, after (IU5), (II.16), and (II.17), half-integer number 8. We denote them by D'. They are (28 + I)-dimensional. U(V)if;(P, E, r) We show now that choosing a paraboloid (11.7') U(VP'E,)U(V~PE),V)U-l(VPE)if;Cp, E, r) (which fixes the possible representations of the translation group) and a representation D' of the U(VP'E,)U(V~PE),V)if;(po, Eo, r) little group completely determines a representation L U(Vp.E,)if;Cpo, Eo, mD·(V~DB).V)]H of the whole Galilei group. ~

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U(V)If(p, E, t) resentation [m I '0, s] of the Galilei group. We will now study this correspondence. = 1:, If(p',E',mD'(Vcip,E),V)],r' (II.18) Since we also know the operators which represent Schrodinger Equation and Galilei Group the translation, the decomposition (II.2) of any For the time being, we shall disregard the spin element of the whole group, gives us at once the and consider the Schrodinger equation for a spinless complete solution to our problem: particle: U(b, a, v, R) If(p , E, t) i(alf/at) + (1/2m),11f = '0 If. (III. I) = exp [-ictm)a·v + ia·p' - ibE'] We are only concerned with the free-particle case, X L If(p', E', ~)[D'(V6P.E)(V,R)]H' (II.19) , so that in (III.I), '0 is a constant. We wish to study the invariance properties of the Schrodinger equation where with respect to the Galilei group transformation. (Let us note that most of quantum mechanics p' = Rp + mv, textbooks thoroughly investigate the Lorentz in 2 E' = E + v·Rp + !mv , variance of the Dirac equation but completely and overlook the Galilean invariance of the Schrodinger equation.) VciPE),(V,R) = V;?E'(O, 0, v, R) VpE . We follow the passive point of view. That is, we These representations are clearly irreducible. They look at the same state described by (IILI), III a are unitary with the scalar product: transformed frame of reference defined by x' = Rx vt a + + (1.2) Ccp, If) = dp dE 0[ E - :~ - '0 ] f t' = t + b. x L (p, E, a)lf(p, E, a). (II.20) G = (b, a, v, R) is the Galilean transformation we consider. In the new frame of reference, the Finally, one can verify directly that they are indeed state must be described by some wavefunction If'. projective representations of the universal covering The physical predictions we get from the two descrip group of the Galilei group with the system of factors tions will be identical if and only if the transformed (L8) which we started from. wavefunction at any point differs from the original The transition to the Galilei group itself merely wavefunction at the transformed point by at most introduces the usual sign ambiguity in the case of a phase factor (the density of particles being a half-integers. We have thus obtained the following scalar) : result: If'(x, t) = e-;[(X' ,t ') If(x', t'), (III.2) The physical representations of the Galilei group are characterized by two real numbers where (x', t') depend on (x, t) according to (1.2). m and '0 and an integer or half-integer number s. Obviously, the new wavefunction has to satisfy the We designate them by [m I '0, s] and they are Schrodinger equation given explicitly by (II.19). i(alf' /at) + (l/2m},1If' = '01f'. (III.3) III. PHYSICAL DISCUSSION We may now determine the unknown function f. The interpretation of the preceding results is Using (1.2), we find straightforward. Eq. (II.20) defines the functional a/at = a/at' v·V', space of the representation as the space of square + (III.4) integrable functions on the paraboloid: V = RV'. E - (p2/2m) = '0. (II.7') The Schrodingerequation (III.3) for the new wave This, and the rotation properties of these basis function can be rewritten as an equation in f and If: functions, impel us to establish a one-to-one cor respondence between a (free) particle of mass m, rica/at') + iv· V' + (1/2m},1' - '0] internal energy '0, spin s, and the physical rep- X e-if(zo"O)If(x', t') = 0. (III.5)

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Dropping the primes and expanding, + timv2t' - iC ]ift(X" t') dx' dt'. (III. 13)

We drop the primes and re-order the sum in the exponential: (IIL6) cp'(p, E) = exp [iRp·a - ibRp·v - iEb - iC] ift(x, t) satisfies the Schrodinger equation (III. I), X exp [-i(Rp mv).x iCE v·Rp so that we are left with the two conditions J + + +

2 Vf - mv = D, tmv )t]ift(x, t) dx dt. (III.7) + 2 (of/at) - (1/2m)t.! + mv = D. We write We easily integrate these equations, and get p' = Rp mY, + (III.14) 2 2 f(x, t) = mv·x - !mv t + C, (III.8) E' = E + v·Rp + tmv • where C is a constant. Expressing (p, E) in terms of (p', E'), we obtain We see that, unlike the relativistic case, the phase cp'(p, E) = exp [-imv·a + timbv2 - iC factor cannot be eliminated. The transformation properties of the Schrodinger wavefunctions are then + ia·p' - ibE']cp(p', E'). (IILI5) if/ex, t) = exp [-imv·x' We now choose

2 2 + !imv t' - iC]ift(x', t'), C = -tma·v + tmbv • (III.16) (III.9) x' = Rx + vt + a, Then t' t + b. cp'(p, E) exp [-itmv·a + ia·p' In the momentum space, we deal with wave - ibE']cp(p', E'). (III.17) functions: One sees at once that (III.17), (III.14), and (III.l2) are identical with (II.19) and (II.20); (III.lI') with . (III.12) particle described by the physical representation [m I '0, 8]. This, however, is not a Galilei invariant Let us study the Galilean transformation in the concept. Indeed, in order to understand the meaning momentum space: 7 The concept of equivalence in the case of projective representations is somewhat distinct from the case of true representations. In fact if {V. I and {V r' I are two projective cp'(p, E) = Je-;P'JC+,Etift'(x, t) dx dt representations of the group G, they are said to be equivalent if U r' = VU.V-I holds between operator rays2 (U. is the operator ray generated by Vr, i.e. the set of all operators TV., TEe, H = 1). For the operators themselves, we have (-ip·R-\x' - vt' = Jexp + vb - a) Vr' = .p(r)VVrV-1, where .p(r) is some complex function of modulus 1 on the group. We see that Vr' is indeed a projective representation of the group with a factor system iE(t' - b)] X exp [ -imv·x' w'(r, s) = [.p(r).p(s)/.p(rs)]w(r, s) equivalent to the factor + system wCr, s) of {V.I.

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of 8, we have to define the little group f)(Po, Eo), that is to pick up some particular point (Po, Eo) of the (cp, 1/1) = J t: (p, E, r)f(p, E, r) V • (0,0, m-1p, 1). J t: p E = (III.I8)

Starting from (1.3), we now easily obtain X o(E - ::) dp)E V6P.E) (v.R)

(0,0,0, R), (III.19) = J t: ¢(P, E + '0, r)1/I(p, E + 'O'~'r) with any V = (0, 0, v, R) and (p', E') Yep, E). Consequently, we rewrite (Il.I9) as X o(E - ::) dp dE U(b, a, v, R)f(p, E, r) = J t: ¢(P, E', r)f(P, E', r) = exp [-iima·v + ia·p' - ibE'] X L f(P', E', ~)[D'(R)], f), (II I. 22) p' = Rp + mY,

2 that is to say, U is an isometric operator and :Je'U E' = E + v·Rp + imv • and :Jeo are isomorphic Hilbert spaces. 8 characterizes now exclusively the behavior of our If U(G) is the operator corresponding to G = particle with respect to rotations; it is really its (b, a, v, R) in the [m I '0, 8] representation, we define

intrinsic angular momentum. 1 O(G) = UU(G)U- • Let us notice that the choice (IlLI8) which led (III.23) us to this result amounts to bringing back the Letting now O(G) act upon some function f(P, E) particle at rest by accelerating the initial coordinate of X o, using (IIL2I) and (II.19), we get system, without rotating it around the direction O(G) eib'O Uo(G) , (III.24) of the movement (pure Galilei transformation). = where Uo(G) is the operator corresponding to G, Internal Energy in the representation [m I 0, 8] according to the definition (II.I9), or else

Physically, we are used to saying that, in non 1 relativistic mechanics, we can freely choose the U'U(G) = eib'UU- Uo(G)U, (III.25) origin from which we count the energies. In the where we add a subscript '0 to U(G) in order to particular case of one free particle, this amounts to emphasize the fact that it belongs to the representa saying that the internal energy is completely tion [m I '0, 8J. Obviously then, the representations arbitrary. We would like to rediscover this feature [m I '0, 8] and [m I 0, 8] are equivalent, in the sense from Galilean invariance. This is done quite easily. of projective representations equivalence. 7 In other Let f(p, E, r) be some basis function of the words, for an isolated particle, the internal energy [m I '0, 8J physical representation of the Galilei '0 has no physical significance. group. Let us now define an operator U by: Antiparticles f(p, E, r) = (Uf)(p, E, r) Until now, when looking for a physical interpreta = f(P, E + '0, r), (III.21) tion of our results, we implicitly assumed tha('m, We callX'U the Hilbert spaces of functions 1/I(P, E, r) which we interpreted as the mass, was positive. with the scalar product: Actually, the construction of the first section is

Downloaded 13 Nov 2012 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions GALl LEI GROUP AND NONRELATIVISTIC QU ANTUM MECHANICS 783 valid for any nonzero value of m. What can be said E' = E + v·Rp. of the negative m case? The basis functions of the representation are now Casting a glance at Eq. (II.19), we see at once defined onto the Hcylinder": that if f(p, E, t) transforms according to the [m I '0,8] representation, then p2 = p2 = ct. (E arbitrary). (IV.2)

1//(p, E, t) = f( -p, -E, t) (III.26) But these representations are no longer irreducible. In fact, let Po (lying on the cylinder IV.2) be the transforms according to [-m ,-'0, sl, where by s vector around which the rotations, in the D' rep we mean that fC transforms under rotations by resentation chosen, are diagonal matrices. We call jj' (R), the complex conjugate representation of ~(po) the group of rotations around Po. Ro being such D' (R). But this does not alter the physical interpret a rotation, with an angle IPo, one has ation of 8 as the spin of the particle. We can immediately verify that the operation (IV.3) f -t fC is an antiunitary one and then, that the We once more apply the little group technique. representations [rn I '0, 81 and [-m I -'0, 8J are Starting from any vector p of the cylinder (IV.2), antiunitarily equivalent. we can choose one rotation r p such that If we consider now charged particles, by the replacement (IVA) p-tp - QA, E -tE - Qq;, (III.27) Now, R being some rotation which takes p into p': we see that the above-mentioned antiunitary trans p' = Rp, (IV.5) formation takes a particle of mass m, internal energy we get '0, and charge Q into another particle characterized respectively by (-rn, -'0, -Q). po = r;,lRrppo, i.e., r;,lRrp = R~·R £ ~(Po). Let us remark that a Fourier transformation, That is, any rotation can be written in the form or else direct dealing with the Schrodinger equation in space-time, leads us to the same result with the (IV.6) transformation We next define new functions cb(P, E, p) on our 1/!C(x, t) = iii(x, t). (III.28) cylinder: In other words, if the representation [m I '0, 8] and (IV.7) the charge Q describe some particle, we may describe its antiparticle either by the same representation They transform according to [m I '0, 8J and the charge -Q, or else by the rep U(b, a, v, R)¢(p, E, p) = E exp (iap' - ibE') resentation [-m I -'0, 8J and the same charge Q. • This gives some meaning to the negative m case. X E f(P', E', mD'(R)]n[D'Crp)JiP' We notice, however, that, as it is well-known, • if the same Dirac equation describes particle and Inverting the summations, and using (IV.7), antiparticle, their nonrelativistic description needs two different Schrodinger equations. U(b, a, v, R)¢(P, E, p) = exp (iap' - ibE') X E E 1/!(p', E', mD·(rp'»)tT[D·(R~·R)J.~. IV. THE ZERO-MASS CASE < T We now plainly make m = 0 in the realization Lastly, (III.20) we obtained for the physical representations UCb, a, v, R)cb(p, E, p) [m I '0, 8J of the GaliIei group. We thus get some tru.e representation of the group: = exp [iap' - ibE' + ifX{>o(R, p)]cb(P', E' p). (IV.S) U(O, a, v, R)f(p, E, t) = exp (ia-p' - ibE') We calculate explicitly the function

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Inonu and Wigner.3 They are characterized by an elements of the physical representation (III.20): integer p and a positive number p2. We denote M = -px(ajap) - is, them by I° I P, p I. The very construction of these representations and as we will see later, their Lie u = m(ajap) + p(ajaE), (V.2) algebra and her invariants, designate p as the k = p, r = E, component of the angular momentum along the with obvious notations. Let us notice, as usual, the direction of the linear momentum, i.e., helicity. splitting of the total angular momentum in orbital Added to the fact that we deal here with null and intrinsic (spin) parts. mass states, this suggests a close analogy between It is now easy to see that the realization (V.2) the representations just found and the irreducible fulfills all equations (V.1), except that translations representations of the Poincare group in the zero and pure Galilei transformations no longer commute. mass case. In fact, at least in the case of vanishing P, Instead, we may interpret the representations 10 I P, p} (V.3) (P ~ 0) as describing zero-mass and infinite-speed particles, which are indeed the nonrelativistic limit This is quite natural. We know that we deal in fact of the zero-mass particles. Naturally, there is no with a projective representation. This means that completely consistent interpretation of the rep in (V.2) we obtain a representation of the Lie resentations just found. Nevertheless, we have been algebra of a central extension of the Galilei group, able to give some vague meaning to them. And and no longer of the Galilei group itself. The Lie mainly, they display certain features we usually algebra element of the one-parameter subgroup by think to be characteristic of the relativistic case which the extension is made can be called p.. Here, it (uncoupling of different helicity states) and which, is represented by p. = m. The extension is central, 8 9 so that p. commutes with all other Lie algebra in fact, are latent in the nonrelativistic case. • elements. But it is nontrivial, so that p. appears in V. THE LIE ALGEBRA OF THE GALILEI GROUP some Lie bracket [see (V.3)]. AND THE ROLE OF THE MASS IN NON The enveloping algebra admits the following RELATIVISTIC QUANTUM MECHANICS invariants: 4 Taking the infinitesimal elements of the one 2 2p.r - k = 2mB - p2 = 2m'O, parameter subgroups of the Galilei group (considered 2 2 as a Lie group) and using the group law (1.3), we -(f.J.M + k XU)2 = m S2 = m s(s + 1), (V.4) calculate their commutators and thus obtain the /J. = m. Lie brackets for the Lie algebra of the group. We recover, of course, the characterization of We make the most natural choice for the basis physical representations by [m I '0, s]. There is elements of the algebra: however a rather subtle point we have yet to make r for the time translations, clear. We have seen that physical knowledge as well k.(i = 1, 2, 3) for the space translations, as mathematical considerations on the Galilei group ui(i = 1, 2, 3) for the pure Galilei transformations, physical representations allow us to conclude that, Mi(i = 1,2,3) for the rotations. in fact, the internal energy of an isolated particle is an arbitrary parameter. Precisely we showed that We then have all representations [m I '0, s] and [m I '0', s] are physically equivalent. But we now find '0 as an [Mi' M j ] = Eijkllfk, lUi' u j ] = [ki' k j ] = 0, element of the center of the group algebra. How fiikUk, [k , r] 0, [Mi' Ui] i (V.1) can any equivalence transformation modify this center? The answer is to be found in the fact that [M i, k j ] Eijkkk, lUi' k j ] 0, we deal here with an extension of the Galilei group, [M , r] • i 0, lUi, rJ k i and such an extension as we consider here has not Let us now compute explicitly the infinitesimal a uniquely defined Lie algebra. There is a whole class of algebras, in one-to-one correspondence with 8 It was Wigner 9 who emphasized (in the relativistic case but the same remark is valid here) that a zero-mass system the unlike but equivalent systems of factors of the possesses two polarization states but only if we consider the space reflections, since otherwise they would not be connected projective representation associated with the exten to each other. On the other hand, for the nonzero mass sion. Here, going from some algebra to another systems, proper rotational invariance is sufficient for deducing the (28 + 1) polarization states from anyone among them. equivalent one, we modify precisely the center 9 E. P. Wigner, Rev. Mod. Phys. 29, 255 (1957). element '0, and that one only.

Now the existence within the center of the becomes trivially a direct product and has no more enveloping algebra of the group, of some basis physical consequences. element of this algebra (here J1., the mass operator), Starting from the true representation (IV.9) of involves a most important physical consequence: it the Galilei group, and using the explicit form leads to a superselection rule. 10 'Po(R, p) derived in the Appendix, we obtain the Let us for instance consider a state vector which following representation: results from the superposition of two state vectors a a. P2 having different masses: Ml = -P2 -;- + P3 -;- + tp -+ upa UP2 P pa (V.5) a a. P2 M2 = -P3 -;- + PI -;- + t p -+ where 1/11 and 1/12 respectively belong to the physical UPI upa P pa (V.9) representations [m 1 I '0, s] and [m2 I '0, s] of the Galilei group. We now consider the behavior of M a = -PI -;-a + P2 -;-a + tp. UP2 UPI this composed state under the following series of transformations: a translation a; a pure Galilei u = p(ajaE), k = p, T = E. transformation v; the inverse translation, and the We may now verify that this representation leads inverse Galilei transformation. Within the group we us to the rules (V.l). We also notice the close analogy know that these all commute, whence, between these expressions and those obtained for (0,0, -v, 1)(0, -a, 0, 1)(0,0, v, 1)(0, a, 0, 1) the Poincare-group Lie algebra in the zero-mass case. 12 = (0,0, 0, 1), (V.6) The enveloping algebra invariants and their values i.e., the identical transformation. for the {O I P, pI representation are With respect to some physical representation, k2 = p2 = p2, that series is obviously represented by some phase (V.lO) factor at most. In fact, using (111.20), we find at once M·k = ipp = ipP. U(O, 0, -v, I)U(O, -a, 0, I)U(O, 0, v, 1) This confirms our interpretation of p as the helicity.

ima v x U(O, a, 0, 1) = e- ' • (V.7) VI. DECOMPOSITION OF THE TENSOR PRODUCT OF TWO PHYSICAL REPRESENTATIONS Thus, our compound state becomes im The tensor product of two physical representations 1/1 = 1/11 + 1/12---" 1/1 = e-im,a,vl/'l + e- ,a'V2' (V.8) of the Galilei group,

The superposition principle cannot have any mean (VI. 1) ing for 1/11 and 1/12 if m 1 ~ m2 , since that would mean is still a (projective) representation whose operators that an identical transformation could affect the act onto the square-integrable basis functions norm of any of their compound states. The relative phase of two states having different masses is 1/I(Pl' P2, EI , E2, tl, t2) completely arbitrary. This is known as the "Brag according to mann superselection rule."ll It prevents the exist ence, in nonrelativistic quantum mechanics, of U(b, a, v, R)1/I(Pl, P2, E I , E 2 , SI, S2) states with a mass spectrum, and therefore of = exp m2)a·v unstable particles. [-!i(ml + We see here how the mass plays different parts + ia(pi + pD - ib(Ei + E~)] in relativistic and nonrelativistic quantum theories. X L 1/I(pi, pL Ei, EL ~1' ~2) hE~ The Lie Algebra in the Zero-Mass Case (VI.2) We will deal now really with the Lie algebra of where the Galilei group itself: the central extension p; = Rpi + m;v, (i = 1,2) 10 G. C. Wick, A. S. Wightman, and E. P. Wigner, Phys. 2 Rev. 88, 101 (1952). E~ = E; + V·Rpi + !miv . 11 A. S. Wightman, "Lectures on Relativistic Invariance," in Les Bouches 1960 Summer School Proceedings (Hermann et 12 J. S. Lomont and H. E. Moses, J. Math. Phys. 3, 405 Cie., Paris, 1960), pp. 159-226. (1962).

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These representations are unitary, the scalar product being defined as

= X L4i(p,E,q,E,rl, r2)I/t(p,E,q,E,rl, r2), (VI. 6) (q" I/t) Jdpl dEl dp2 dE2 rd"3 which has been directly derived from (VI.3). X O(EI - 2~1 - '0 1) o(E2 - 2~2 - '02) The Hilbert space X of the representation is thus reduced to a direct integral of Hilbert spaces X"U, X L 4i(Pl' P2, E l , E 2 , rl, r2) ',f. each associated to a paraboloid: (VI.3) (VI. 7) We are going to reduce the Hilbert space, which Since (P, E) precisely characterizes the representa we just determined, into a direct sum (possibly a tion of the translation subgroup in the tensor direct integral) of invariant Hilbert spaces. We product (see VI.I), the condition (VI.7) is then a proceed quite similarly to Wightman.ll necessary one for the Hilbert spaces X"U to be We first define new variables: invariant. In fact, we can reduce them no further P = PI + P2, E = El + E2 , with respect to the translation subgroup representa tion, and it suffices now to look in each X"U for the E = (VIA) subspaces invariant with respect to the little group, which.may be chosen simply as the rotation group. and call Equation (VI.6) shows now that, if some point mlm2 (P, E) is fixed, the basis functions of our representa p.= m l + m2 ' tion only depend on some vector q whose length is fixed. Since these are uniform functions onto where we have assumed m m2 ~ 1 + o. the sphere q2 = C", we expand them in spherical These variables are precisely those corresponding harmonics, i.e., basis functions of irreducible rep to the usual separation of the center-of-mass and resentations of the rotation group. We then obtain, relative motions for our two-particle system. for the little group (rotation group), the representa We also introduce the internal energy of the tion compound system, i.e., the difference between its total energy and the center-of-mass kinetic energy: I D" ® D" ® (ffi D ). (VI.8) (VI. 5) 1-0 We have now the following expression for the The decomposition of this tensor product into scalar product: irreducible representations is immediate. Since we have seen at the beginning that a paraboloid '() and (q" I/t) = 1'" d'O JdP dE o(E - ~ - '0) an irreducible representation of the little group "U,+"U. 2M uniquely define a physical representation of the Galilei group, we finally obtain the complete solution to our problem, which we symbolically write as:

We see, exactly as in the case of the Poincare superconserved as we have seen). The kinetic energy group,11 the most natural appearance of the orbital of the relative motion of the components is to be angular momenta. found now in the internal energy of the compound This provides also the profound reason why, when system. The weakened concept of equivalence which studying the Schrodinger equation, one keeps only we introduced earlier and which led us to the integer (and not half-integer) relative angular arbitrariness of the internal energy of an isolated momenta. But in contradistinction with the rela particle, has now also, in the two-particle case, a tivistic case, the mass is now conserved (and even quite interesting application. It enables us to change

[ml I '01, 8tl and [m2 I '02, 82] into [ml I 0, 81] and interest they have taken in this work, as well as [m2 I 0, 82], but not to change simultaneously all for their many suggestions and critical remarks. the [m l + m, / '0, j] of their tensor product de composition into [m l + ma I 0, jJ. All we can do APPENDIX is to "renormalize" the internal energy of the possible We here derive an explicit formula for the angle compound states by an amount '0 1 + '02 , Of course this agrees entirely with our previous physical CPo of the rotation: knowledge: once we have fixed up the internal (A.I) energy of two isolated particles, their compound state has an internal energy which is no longer where p' = Rp, and the r/s are well defined rotations arbitrary. which take some fixed Po into p:

In the case where m l + ma = 0, with the help rppo = p. of techniques quite similar to those just used, we obtain the following result, which we quote here We use the spinorial representation of the rotation for completeness and without proof: group. A rotation by angle around an axis n(/n/ = 1) may be written as [m 1'01 ,811@ [-m / '02 , 82 1 R(n, cp) = cos (cp/2) - itt sin (cp/2) , (A.2) dP (0 P, PI P2 l}, = J$: l~m P.(jj.. P,~:. I + + where (VI. 10) or else n = ~·n, (A.3)

[m / '01 , sd @ [-m / '02 , S21 and the To'S are the usual Pauli matrices. We choose as rp , the rotation in the (Po, p) plane = 1'" dP EB (0 I P, pj0(2,,+I) (2 •• +1). (VI. 10') which brings Po into p. Writing it as the product $0 p __ CXl of two plane symmetries, we have Such a result is valid also if we replace one (or two) of the representations [m / '0, s] by [m / '0, sl (see (A A) Sec. III); particularly, where k and ko are the unit vectors lying on p [m / '0, sl @ [-m / '0', sl and PoCk = pi/pi, etc .... ), and with the same notations as in (A.3). = 1" dP EB (O I P, p}0(2Hl)', (VI. 11) eo p_-m Now, from the definition of k' and (A.2), we have and this justifies our choosing the representa tions [m / '0, s] and [-m / '0', s] in order to represent, k= (cos ~ - in sin ~)k( cos ~ + itt sin ~) . (A.5) respectively, a particle and its antiparticle. All the results derived here are of course well Bringing (A.S), (AA), and (A.2) into (A.I), using known. It is however stimulating to obtain them also, repeatedly, the well-known identity from Galilean invariance only, and this provides an db = a·b + i~·(a x b), (A.6) agreeable and unifying piont of view. ACKNOWLEDGMENTS and The author is very grateful to Professor Louis Michel and Professor Franc;ois Lur<;at for the we get

(2 + feko + feofe) cos ~ - i(tt + kottko + ttkko + feottfe o) sin i Ro = 2(1 + ko ·k')I(I + ko .k)! Finally, R _ (1 + k·ko) cos (cp/2) + (ko, n, k) sin (cp/2) - iko(n·ko + n·k) sin (cp/2) . (A.7) o - (1 + k.ko)t[I + cos cpko·k + (1 - cos cp)(n·ko)(n·k) + sin cp(ko, n, k)]l

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The expression (A.7) for Ro shows clearly the axis Choosing ko as our z axis, and taking successively ko of the rotation R o, whose angle CPo is immediately infinitesimal rotations around the x, y, and z axes, obtained by identifying (A.7), and we obtain (cpo)x = [kz/O + k,]cp = [Px/(p + p,)]cp, Ro = cos (CPo/2) - ifco sin (CPo/2). (A.7') (CPo). = [k yl(l + k,)]cp = [P./(p + pz)]cp, (A.9) In particular, for an infinitesimal rotation around (CPo), = [1 + k,/(l + k,)Jcp = cpo n, with an angle of cP « 1, We can then immediately derive the expressions CPo ~ (n·ko + n·k/1 + k·ko)cp. (A.8 (V.9) for the Lie algebra in the zero-mass case.

JOURNAL OF MATHEMATICAL PHYSICS VOLUME 4. NUMBER 6 JUNE 1963

Principle of General Q Covariance

D. FINKELSTEIN,* Yeshiva University, New York, New York J. M. JAUCH, University of Geneva and CERN, Geneva, Switzerland S. SCHIMINOVICH, t Yeshiva University, New York, New York AND D. SPEISER,t University of Geneva, Geneva, Switzerland (Received 10 December 1962)

In this paper the physical implications of quaternion quantum mechanics are further explored. In a quanternionic Hilbert space Xo, the lattice of subspaces has a symmetry group which is iso morphic to the group of all co-unitary transformations in Xo. In contrast to the complex space Xc (ordinary Hilbert space), this group is connected, while for Xc it consists of two disconnected pieces. The subgroup of transformations in Xo which associates with every quaternion q of magnitude 1, the correspondence if/ ..... qif/q-l for all if/ E X Q (called Q conjugations), is isomorphic to the three dimensional rotation group. We postulate the principle of Q covariance: The physical laws are in variant under Q conjugations. The full significance of this postulate is brought to light in localizable systems where it can be generalized to the principle of general Q covariance: Physical laws are in variant under general Q conjugations. Under the latter we understand conjugation transformations which vary continuously from point to point. The implementation of this principle forces us to construct a theory of parallel transport of quater nions. The notions of Q-covariant derivative and Q curvature are natural consequences thereof. There is a further new structure built into the quaternionic frame through the equations of motion. These equations single out a purely imaginary quaternion "I(x) which may be a continuous function of the space-time coordinates. It corresponds to the i in the Schriidinger equation of ordinary quantum mechanics. We consider "I(x) as a fundamental field, much like the tensor gp.. in the general theory of relativity. We give here a classical theory of this field by assuming the simplest invariant Lagrangian which can be constructed out of "I and the covariant Q connection. It is shown that this theory describes three vector fields, two of them with mass and charge, and one massless and neutral. The latter is identifiable with the classical electromagnetic field.

1. INTRODUCTION approximate validity and that the true laws of N the development from Galilean to special to geometry are subject to disturbances from place to I general relativity, it was shown by Einstein that place. Still more fundamental than the laws of the concepts of Euclidean geometry have only an geometry are those of classical logic as expressed in the propositional calculus. In the development * Supported by the National Science Foundation. t Now at the University of Buenos Aires. from classical to quantum physics it was shown t Supported by the Swiss Commision for Atomic Research. by Bohr that the concepts of classical logic have

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