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

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REPRESENTATIONS of the POINCARE GROUP for QUANTUM FIELD THEORY by James Kettner REPRESENTATIONS OF THE POINCARE GROUP FOR QUANTUM FIELD THEORY by James Kettner The uni…cation of quantum mechanics and special relativity into quantum …eld theory still contains some of the major assumptions of non-relativistic quan- tum mechanics. In particular, it is still postulated that a physical state corre- sponds to a vector in a Hilbert space up to multiplication by a complex number, i.e. states correspond to rays in a Hilbert space. The probability of a transition from a state represented by a ray = ; where is a vector in the Hilbert f g space and = 0 is a complex number, to a state represented by = is given by the6 ray product b f g b (; 2 [ ] = j j j (; )( ; ) where (; ) is the inner productb b on the Hilbert space; this de…nition does not depend on the choice of vectors in and in . Therefore the transfor- mations that preserve probabilities are only de…ned up to a complex factor. Everybody can safely ignore this complicationb because,b in 1931, Eugene Wigner proved that any transformation preserving probabilities can be replaced with either a linear, unitary transformation or an anti-linear, anti-unitary transfor- mation. In 1939, Wigner published a paper with important results relating quantum mechanics and special relativity. These results are: 1. A Hilbert space of states carries a representation, up to a factor, of the inhomogeneous Lorentz group by linear, unitary operators. Wigner states that "... there corresponds to every invariant quantum mechanical system of equations such a representation of the inhomogeneous Lorentz group. This representation, on the other hand, though not su¢ cient to replace quantum mechanical operators entirely, can replace them to a large ex- tent." 2. 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 inho- mogeneous 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 inhomoge- neous 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 iden- tity 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 representa- tion 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 quan- tum mechanics.
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