Wigner's theorem

The theorem specifies how physical symmetries such as rotations, translations, and CPT transformations are represented on the Hilbert space of states.The physical states in a quantum theory are represented by unit vectors in Hilbert space up to a phase factor, i.e. by the complex line or ray the vector spans.In addition, by the Born rule the absolute value of the unit vector's inner product with a unit eigenvector, or equivalently the cosine squared of the angle between the lines the vectors span, corresponds to the transition probability.By Wigner's theorem, any transformation of ray space that preserves the absolute value of the inner products can be represented by a unitary or antiunitary transformation of Hilbert space, which is unique up to a phase factor.It is a postulate of quantum mechanics that state vectors in complex separable Hilbert spacethat are scalar nonzero multiples of each other represent the same pure state, i.e., the vectorsdefine the same ray, if and only if they differ by some nonzero complex number:as a set of vectors with norm 1, a unit ray, by intersecting the lineby and define ray space as the quotient set Alternatively, for an equivalence relation on the sphereContrarily to the case of vector spaces, however, an independent spanning set does not suffice for defining coordinates (see: projective frame).Note that the righthand side is independent of the choice of representatives.The physical significance of this definition is that according to the Born rule, another postulate of quantum mechanics, the transition probabilities between normalised statesThis becomes even clearer when one considers the mathematically equivalent passive transformations, i.e. simply changes of coordinates and let the system be.An exception would be (in a non-relativistic theory) the Hilbert space of electron states that is subjected to a charge conjugation transformation.However this means that the symmetry acts on the direct sum of the Hilbert spaces.A geometric interpretation is that a symmetry transformation is an isometry of ray space.reduces to a unitary operator whose inverse is equal to its adjointmust be represented by a unitary or antiunitary operator is determined by topology.Remark 3: Wigner's theorem is in close connection with the fundamental theorem of projective geometry[15] If G is a symmetry group (in this latter sense of being embedded as a subgroup of the symmetry group of the system acting on ray space), and if f, g, h ∈ G with fg = h, then where the T are ray transformations.From the uniqueness part of Wigner's theorem, one has for the compatible representatives U, where ω(f, g) is a phase factor.If the realization of the symmetry group, g → T(g), is given in terms of action on the space of unit rays S = PH, then it is a projective representation G → PGL(H) in the mathematical sense, while its representative on Hilbert space is a projective representation G → GL(H) in the physical sense.Applying the last relation (several times) to the product fgh and appealing to the known associativity of multiplication of operators on H, one finds They also satisfy Upon redefinition of the phases, which is allowed by last theorem, one finds[16][17] where the hatted quantities are defined by The following rather technical theorems and many more can be found, with accessible proofs, in Bargmann (1954).For their respective universal covering groups, SL(2,C) and Spin(3), it is according to the theorem possible to have ω(g, h) = 1, i.e. they are proper representations.Two functions related as the hatted and non-hatted versions of ω above are said to be cohomologous.[19][20] Assuming the projective representation g → T(g) is weakly continuous, two relevant theorems can be stated.An immediate consequence of (weak) continuity is that the identity component is represented by unitary operators.[nb 4] Theorem: (Wigner 1939) — The phase freedom can be used such that in a some neighborhood of the identity the map g → U(g) is strongly continuous.More precisely, this is exactly the case when the second cohomology group H2(g, R) of the Lie algebra g of G is trivial.[21] Wigner's theorem applies to automorphisms on the Hilbert space of pure states.Theorems by Kadison[22] and Simon[23] apply to the space of mixed states (trace-class positive operators) and use slight different notions of symmetry.