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In elementary wave mechanics, the overall phase of a wave-function is not observable. In general quantum mechanics, this idea leads to the postulate that given a vector Ψ in Hilbert space, all vectors differing from by a complex non-zero multiple (=the ray containing should represent the same state of the system. Geometrically, we say that the relevant space is the set of rays, known as the projective Hilbert space. The interpretation of the scalar product in terms of probability means that, by convention, we need consider only rays of unit length, so Wigner starts with the set of unit rays. Note that the rays do not themselves form a linear space. A vector in a given unit ray might be used to represent the physical state more conveniently than itself, but is ambiguous up to a phase (complex multiple of unit modulus). The transition probability between two rays and can be defined in terms of vector representatives and to be
and is independent of which representative vectors, and , are chosen. Wigner postulated that for the transition probability between states to be the same to all observers related by a transformation of special relativity. More generally, he considered the statement that a theory be invariant under a group G to be expressed in terms of the invariance of the transition probability between any two rays. The statement postulates that the group acts on the set of rays, that is, on projective space. Let (a,L) be an element of the Poincaré group (the inhomogeneous Lorentz group). Thus, a is a real Lorentz four-vector representing the change of space-time origin
and L is a Lorentz transformationThe Lorentz transformation (LT), named after its discoverer, the Dutch physicist and mathematician Hendrik Antoon Lorentz ( 1853- 1928), forms the basis for the special theory of relativity, which has been introduced to remove contradictions between the t, which can be defined as a linear transformation of four-dimensional space-time which preserves the Lorentz distance of every vector . Then the theory is invariant under the Poincare group if for every ray Ψ of the Hilbert space and every group element (a,L) is given a transformed ray Ψ(a,L) and the transition probability is unchanged by the transformation:
The first theorem of Wigner is that under these conditions, we can express invariance more conveniently in terms of linear or anti-linear operators (indeed, unitaryIn government, see Unitary state In mathematics, see Unitary matrix Unitary operator Unitary group Unitary representation. or antiunitary operators); the symmetry operator on the projective space of rays can be lifted to the underlying Hilbert space. This being done for each group element (a,L), we get a family of unitary or antiunitary operatore U(a,L) on our Hilbert space, such that the ray Psi transformed by (a,L) is the same as the ray containing U(a,L)psi. If we restrict attention to elements of the group connected to the identity, then the anti-unitary case does not occur. Let (a,L) and (b,M) be two Poincare transformations, and let us denote their group product by (a,L).(b,M); from the physical interpretation we see that the ray containing U(a,L)[U(b,M)]psi must (for any psi) be the ray containing U((a,L).(b,M))psi. Therefore these two vectors differ by a phase, which can depend on the two group elements (a,L) and (b,M). These two vectors do not need to be equal, however. Indeed, for particles of spin 1/2, they cannot be equal for all group elements. By further use of arbitrary phase-changes, Wigner showed that the product of the representing unitary operatorsIn mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π g is a unitary operator for every g ∈ G''. The general theory is well-developed in case G is a locally compact (Hau obeys
instead of the group law. For particles of integer spin (pions, photons, gravitons...) one can remove the +/- sign by further phase changes, but for representations of half-odd-spin, we cannot, and that the sign changes discontinuously as we go round any axis by an angle of 2 pi. We can, however, construct a representation of the covering group of the Poincare group, called the inhomogeneous SL(2,C); this has elements (a,A) where as before, a is a four-vector, but now A is a complex 2 times 2 matrix with unit determinant. We denote the unitary operatorIn functional analysis, a unitary operator is a bounded linear operator U on a Hilbert space satisfying U U UU I where I is the identity operator. This property is equivalent to any of the following: U is a surjective isometry U preserves the inner producs we get by U(a, A), and these give us a continuous, unitary and true representation in that the collection of U(a,A) obey the group law of the inhomogeneous SL(2,C).
Because of the sign-change under rotations by 2 pi, Hermitian operators transforming as spin 1/2, 3/2 etc cannot be observableIn physics, particularly in quantum physics a system observable is a property of the system state that can be determined by some sequence of physical operations. These operations might involve submitting the system to various electromagnetic fields and evs. This shows up as the univalence superselection rule: phases between states of spin 0,1,2 etc and those of spin 1/2,3/2 etc., are not observable. This rule is in addition to the non-observability of the overall phase of a state vector. Concerning the observables, and states |v), we get a representation U(a,L) of Poincaré group, on integer spin subspaces, and U(a,A) of the inhomogeneous SL(2,C) on half-odd-integer subspaces, which acts according to the following interpretation:
An ensemble corresponding to U(a,L)|v) is to be interpreted with respect to the coordinates in exactly the same way as an ensemble corresponding to |v) is interpreted with respect to the coordinates x; and similarly for the odd subspaces.
The group of space-time translations is commutative, and so the operators can be simultaneously diagonalised. The generators of these groups give us four self-adjoint operators, , j=1,2,3, which transform under the homogeneous group as a four-vector, called the energy-momentum four-vector.
The second part of the zeroth axiom of Wightman is that the representation U(a,A) fulfills the spectral condition - that the simultaneous spectrum of energy-momentum is contained in the forward cone: