REPRESENTATIONS of the POINCARE GROUP for QUANTUM FIELD THEORY by James Kettner

REPRESENTATIONS OF THE POINCARE GROUP FOR QUANTUM FIELD THEORY by James Kettner

The uni…cation of quantum mechanics and special relativity into quantum …eld theory still contains some of the major assumptions of non-relativistic quantum mechanics. In particular, it is still postulated that a physical state corresponds to a vector in a Hilbert space up to multiplication by a complex number, i.e. states correspond to rays in a Hilbert space. The probability of a transition from a state represented by a ray = ; where is a vector in the Hilbert f g space and = 0 is a complex number, to a state represented by = is given by the6 ray product b f g b (; 2 [ ] = j j j (; )( ; ) where (; ) is the inner productb b on the Hilbert space; this de…nition does not depend on the choice of vectors in and in . Therefore the transformations that preserve probabilities are only de…ned up to a complex factor. Everybody can safely ignore this complicationb because,b in 1931, Eugene Wigner proved that any transformation preserving probabilities can be replaced with either a linear, unitary transformation or an anti-linear, anti-unitary transformation. In 1939, Wigner published a paper with important results relating quantum mechanics and special relativity. These results are:

1. A Hilbert space of states carries a representation, up to a factor, of the inhomogeneous Lorentz group by linear, unitary operators. Wigner states that "... there corresponds to every invariant quantum mechanical system of equations such a representation of the inhomogeneous Lorentz group. This representation, on the other hand, though not su¢ cient to replace quantum mechanical operators entirely, can replace them to a large ex- tent." 2. Any ray representation of the Poincare group can, by a suitable choice of phases, be made into an ordinary representation of the covering group. 3. The determination of all unitary, irreducible representations of the inhomogeneous Lorentz group.

The last result is of the most interest because the representations of the …rst result can be built from the irreducible representations. We suspect that an elementary particle "is" an irreducible unitary representation of the inhomogeneous Lorentz group, the semi-direct product of the translations of Minkowski space R1;3and SO+(1; 3). The representation satisfying the physical conditions can be parameterized by two parameters m and s, where m is a real number

1 (mass) and is constrained by: 1 3 If m2 > 0, s = 0; ; 1; ; ::: (spin), and 2 2 1 3 if m2 = 0, s = 0; ; 1; ; ::: (helicity). 2 2 The purpose of this note is to derive this result.

1 Preliminaries

Minkowski space or spacetime, R1;3, is the four dimensional real vector space with the metric = diag(1; 1; 1; 1). The Lorentz group O(1; 3) is the set of all linear transformations : R1;3 R1;3 such that T = . The Lorentz group has four connected components,! but the component of the identity transformation (the proper orthochronous Lorentz group of transformations 0 with det = 1 and 0 1) is in 1-1 correspondence with each of the other three components via the distinct symmetries of time reversal T, parity P, and their product PT. Henceforth, when we refer to the Lorentz group, we mean the proper orthochronous Lorentz group, SO+(1; 3). If we add the translations by a 4-vector in R1;3 to the Lorentz group, we get the Poincare group P with elements (a; ), where a is in R1;3 and is in SO+(1; 3). The multiplication is

(a; ) (b; 0) = (a + b; 0) 1 1 The identity is (0;I) and the inverse of (a; ) is a; . P is thus the + semi-direct product of the abelian group of translations and SO (1; 3). Now, looking at the translations (b; I) as a subgroup of P, it is invariant (normal) because (a; ) (b; I) = (a + b; ) = (b; I)(a; ). There is a 1-1, onto function M from Minkowski space to the set of 2 by 2 Hermitian matrices with real diagonal entries: x0 + x3 x1 ix2 M(x) = x = x1 + ix2 x0 x3 where the are the Pauli matrices. Clearly M(x)y = M(x). Also det(M(x)) = 0 2 1 2 2 2 3 2 2 (x ) (x ) (x ) (x ) = x ; then, given A in SL(2; C), det(AM(x)Ay) = 2 j j det(M(x)) = x . This action indirectly de…nes a homomorphism S : SL(2; C) + j j ! SO (1; 3) by M(S(A)x) = AM(x)Ay since M is 1-1. S is onto because M is onto.

M(S(A)S(b)x) = AM(S(B)x)Ay

= AB(M(x))ByAy = M(S(AB)x) shows that S is a homomorphism. S is two to one because S(A) = S( A). The matrix elements of S(A) are given by: 1 S(A) = tr( A A ) 2 y

2 SL(2; C) is di¤eomorphic to SO+(1; 3) in a neighborhood of the identity so they have isomorphic Lie algebras. SL(2; C) is actually the universal covering group of SO+(1; 3). The Lie algebra of the Poincare group has generators P = i@ and L = i (x@ x @) with

[P;P ] = 0

[P;L] = i( P P) [L ;L] = i L L + L L . 1 An alternative presentation can be made by de…ning Ji "ijkLjk and Ki 2 L0i; then

[P;P ] = 0

[P0;Ji] = 0 [Pi;Jj] = i"ijkPk

[P0;Ki] = iPi [Pi;Kj] = iijP0

[Ji;Jj] = i"ijkJk [Ki;Kj] = i"ijkJk [Ji;Kk] = i"ijkKk

Dropping the momentum operators, we have a representation of so(1; 3). An- other useful presentation of so(1; 3) is obtained by de…ning M 1 (J + iK ) i p2 i i 1 and Ni (Ji iKi) which yields p2

[Mi;Mj] = i"ijkMk

[Ni;Nj] = i"ijkNk

[Mi;Nj] = 0 which shows so(1; 3) is isomorphic to su(2) su(2). The representation theory of su(2) is well-known from ordinary quantum mechanics as the theory of angular momentum. The representations obtained using this isomorphism are labeled by j and j with (2j + 1)(2j0 + 1) degrees 0 of freedom where 2j; 2j0 = 0; 1; 2; :::. Note, however, that these are not unitary representations over the Lorentz group because

Ki = i (Mi Ni) cannot be Hermitian and they are not irreducible. In fact there is no representation of the Lorentz group or the Poincare group that is both …nite dimensional and unitary because both groups are not compact. To incorporate special relativity, the space of states must be a Hilbert space of functions on MInkowski space that is a unitary representation of the Poincare group. Since the momentum operators all commute, we can label states by their 4-momentum p and a symbol representing any other parameters we need to

3 specify the state completely: the state is p; in Dirac notation. U(a; ) is the linear transformation of the state spacej ini the representation. Since the momentum operators must have P p; = p p; , the action of translations is given by j i j i ip a U(a; I) = e p; j i The action of a Lorentz transformation must take a state with momentum p to one with momentum p so U(0; ) p; is a linear combination of the states j i p; 0 :

U(0; ) p; = C (; p; ) p; 0 j i 0 X0 If it is possible to choose the quantum numbers so that C0 is block diagonal, each block is a representation. If the process cannot be repeated within a block, the representation is irreducible.

2 Induced Representations

We know the action of translations, so we need to …nd the action of Lorentz transformations. The idea is to …x a momentum k and to …nd the orbit of k under the Lorentz group; any vector in that orbit can be written p = (p; k)k where (p; k) is a Lorentz transformation taking k to p, but it is not unique because we can include any Lorentz transformation …xing k. The set of Lorentz transformations …xing k is a group that Wigner calls the little group. Given a unitary representation of the little group by D,

D(`) k; = D(`) k; j i j i X select a speci…c 0(p; k) for each p in the orbit of k. De…ne state vectors

p; U(0(p; k)) k; j i j i where on the left is de…ned to be the same as on the right. To keep the notation as uncluttered as possible, we will drop the U. Under an arbitrary Lorentz transformation

(p0; p) p; = (p0; p)0(p; k) k; j i 1 j i = 0(p0; k) (p0; k)(p0; p)0(p; k) k; 0 j i but the bracketed terms take k to p to p0 then back to k so it is an element ` of the little group called the Wigner rotation. Then

(p0; p) p; = 0(p0; k) D(`) k; j i j i X = D(`)0(p0; k) k; j i X D(`) p0; j i X 4 showing that the states p; transform in a unitary representation of the Poincare group induced fromj i the unitary representation of the little group. Weinberg shows that there is a choice of the 0(p; k) such that, when (p0; p) is a rotation, then the Wigner rotation is identical to (p0; p), i.e. massive particles in quantum …eld theory transform under a rotation as in non-relativistic quantum mechanics. Thus we retain tools of non-relativistic quantum mechanics relating to rotations, e.g. spherical harmonics and Clebsch-Gordan coe¢ cients.

3 Orbits and Little Groups

The orbits of the Lorentz group and generic elements of each orbit are given in the table: Class Orbit Description Generic element k 2 2 0 (m+) p = m > 0; p > 0 Hyperboloid in forward cone (m; 0; 0; 0) (m ) p2 = m2 > 0; p0 < 0 Hyperboloid in backward cone ( m; 0; 0; 0) 2 0 (0+) p = 0; p > 0 Surface of forward cone (; 0; 0; ) (0 ) p2 = 0; p0 < 0 Surface of backward cone ( ; 0; 0; ) () p2 = 2 Space-like hyperboloid (0; 0; 0; ) (00) p = 0 Point (0; 0; 0; 0) We now compute the little group and Lie algebra for each orbit. Since the Lie algebra is the tangent space at the identity of the little group, we can get the generators in SO+(1; 3) by selecting the parameters so that we have a path in the little group which is the identity at 0, di¤erentiating, evaluating at 0 and multiplying by i (to make them Hermitian). 3.1 (m ): M(q) = mI so, if AM(q)Ay = M(q), then AAy = I so the little group is SU(2) with generators J1, J2, and J3. The irreducible unitary representations 1 3 of su(2) are 2j + 1 dimensional where j = 0; 2 ; 1; 2 ; ::: with eigenstates labeled 2 2 2 2 by the eigenvalues of J = J1 + J2 + J3 and J3 which are j(j + 1) and m going by integer steps from j to j. Recall that result comes from de…ning ladder operators J = J1 iJ2 which raise and lower the eigenvalue of J3 by 2 2 1. There is a maximum eigenvalue since J3 J . The constructions are purely algebraic. The matrix elements are

j; m0 J3 j; m = mm m h j j i 0 j; m0 J j; m = (j m + 1)(j m)m ;m+1 h j j i 0 p 3.2 (0 ): 2 2 0 a b 2 0 a c 2 a 2ac M(q) = so, if = j j 2 = 0 0 c d 0 0 b d 2ac 2 c j j 2 0 2 , then a = 1 and c = 0. Therefore the matrices of the little group 0 0 j j

5 ei=2 b are of the form (the division by 2 simpli…es calculations) which 0 e i=2 is just the semidirect product of the abelian group of translations of R2 and rotations within the plane. In the Lorentz group, they take the form

1 2 1 i=2 i i=2 1 2 1 + 2 b 2 (e b + cc) 2 (e b cc) 2 b 1 i=2 j j 1 i=j2 j 2 (e b + cc) cos sin 2 (e b + cc) 0 i i=2 i i=2 1 2 e b cc sin cos 2 e b cc 2 2 B 1 b 1 (ei=2b + cc) i e i=2b cc 1 1 b C B 2 2 2 2 C @ j j j j A To get the tangent space at the identity, take paths through the identity varying the parameters one at a time: for , we get J3; for the real part of b, we have J2 + K1 M; and for the imaginary part of b, we have J1 K2 N . Their commutators are:

[J3;M] = iN [J3;N] = iM [M;N] = 0

We can proceed in much the same manner as we did for SU(2). De…ne ladder operators L = N iM; then [J3;L ] = L . However, each eigenvalue of J3 gives rise to an in…nite tower of eigenvalues which di¤er by integers because 2 2 there is nothing similar to J3 J . Weinberg points out the possibility of a continuum of spins and rules it out by stating "Massless particles are not observed to have any continuous degree of freedom." To rule them out, states must be eigenvectors of M and N with zero eigenvalues. However some papers have been written about the possibility of continuous spin in string theory. We now restrict the discussion to discrete states. We still do not know the eigenvalues of J3. We will try to address that in the last section.

3.3 (): 0 i 0 1 0 1 M(q) = . A is in the little group if A …xes , but A AT = i 0 1 0 1 0 0 ad + bc 0 1 0 1 0 1 = which means A A = A AT ad bc 0 1 0 1 0 y 1 0 which is only possible if A = A, i.e. A has only real entries. Therefore the little group is SL(2; R). In the Lorentz group, they take the form

1 2 2 2 2 1 2 2 2 2 2 x + y + z + w xy + zw 0 2 x y + z w xz + yw xw + yz 0 xz yw 0 0 0 1 0 1 B 1 x2 + y2 z2 w2 xy zw 0 1 x2 y2 z2 + w2 C B 2 2 C @ A

6 x y 1+yz where is in SL(2; R). Since xw yz = 1, w = near the identity. z w x The general form of a tangent vector at the identity is

0 y0(0) + z0(0) 0 2x0(0) y0(0) + z0(0) 0 0 z0(0) y0(0) 0 0 0 0 0 1 B 2x0(0) (z0(0) y0(0)) 0 0 C B C @ A which gives a basis consisting of K3 (choose the constant path at 0 for y and z); K1 (choose identical y and z paths); and J2 (choose y to be the reverse of z).

3.4 (0):

The little group is all of SL(2; C).

4 Massive representations

The little group is SU(2) which has irreducible unitary representations of dimen- 1 3 sion 2j+1 for each j = 0; 2 ; 1; 2 ; :::. States are speci…ed by p; j; m where p is the momentum, j comes from the eigenvalue j(j + 1) of J 2, andj m =i j; j + 1; :::j is the eigenvalue of J3. For …xed j, this representation is irreducible. It is unitary by construction and in…nite because !p is unbounded. The actual matrix elements of the representation are calculated in Tung.

5 Mass 0 representations

The little group is the group of Euclidean rotations and translations with generators J3;P1;P2 as found above. The states are labelled p; where is the j i eigenvalue of J3; any two eigenvalues di¤er by an integer. Since our represen- tative vector had 3-momentum in the 3-direction, represents the component of the angular momentum in the direction of motion, i.e. helicity. Again the matrix elements of the the representation are in Tung. The eigenvalues of J3 are restricted in value by the topology of the Poincare group. SL(2; C) is homeomorphic to R3 SU(2) since there are three indepen- dent boost parameters that take on all real values. A unitary matrix can be x y written in the form with x 2 + y 2 = 1 which describes the 3-sphere y x j j j j S3 in R4 ; S3 is simply connected (any loop can be continuously deformed to a loop constant at a point) because a continuous map from S1 to S3 cannot be onto and we can use stereographic projection form a missed point to the plane (R4) of its equator where it can continuously deformed to a point and the inverse stereographic projection will continuously deform it to a point on S3. The Lorentz group is homeomorphic to R3 SO(3) using the reasoning we used for SL(2; C). Now SO(3) is homeomorphic

7 Figure1, The top left shows a path contractible to 0 while the top right shows one that cannot be contracted to 0. The bottom left shows a path that loops twice; in the middle it has been deformed to separate the branches at their meeting with the boundary; in the right, it has been deformed so it does not cross itself at the origin. Clearly you continuously deform the path on the boundary so that Q1 = Q2 (and therefore Q10 = Q20 ) to get the picture on the top left which is contractible to 0. (Adapted from Sattinger and Weaver).

to a ball of radius with the antipodal points on the boundary identi…ed because a rotation is speci…ed by a direction (given by the angles in spherical coordinates) and magnitude (the radius); the identi…cations are necessary because the rotation through in one direction is equal to the rotation through in the opposite direction. The loop properties of SO(3) can be illustrated in Figure 1. Since SO+(1; 3) is not simply connected, it does have intrinsic ray repre- i sentations (i.e. representations where U()U(0) = e U(0)). Take paths

8 de…ned on the closed interval between 0 and 1 in SO+(1; 3): (s) and (s) such that (0) = I = (0), (1) = , and (1) = ; then (s)(s) is a path in the Lorentz group from the identity to . Then the path in the representation 1 U given by U((s))U((s))U ((s)) is a path from I to I (since following the path twice is contractible, the phase change following it once must be 1). A rotation of 2 around the 3-axis produces a phase change of e2i for the helicity eigenstate. Since it must be 1, must be half-integral. We have been assuming the topological result that loop multiplication in a topological group (follow one loop, then follow the next) is homotopic to point by point multiplication of the loops in the group and the fact as shown in Weinberg that the phase change depends only on the homotopy class of the loop.

6 BIBLIOGRAPHY

1. Haag, R., Local Quantum Physics, Springer, New York (1996). 2. Jones, H. F., Groups, Representations and Physics, IOP, Philadelphia (1998). 3. Sattinger, D.H., Weaver, O.L., Lie Groups with Applications to Physics, Geometry, and Mechanics, Springer-Verlag, New York (1986). 4. Sternberg, S., Group Theory and Physics, Cambridge University Press, New York (1995). 5. Tung, Wu-Ki, Group Theory in Physics, World Scienti…c, New Jersey (1985). 6. Weinberg, S., The Quantum Theory of Fields, Cambridge University Press, New York (1995). 7. Wigner, E., "On Unitary Representations of the Inhomogeneous Lorentz Group," Annals of Mathematics 40, 149-204 (1939). 8. Wigner, E., Gruppentheorie und ihre Anwendung auf die Qunatenmechanik der Atomspektren, Braunschweig (1931).