Wigner's theorem
Wigner's theorem, proved by Eugene Wigner in 1931, is a cornerstone of the mathematical formulation of quantum mechanics. The theorem specifies how physical symmetries such as rotations, translations, and CPT 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 vectors inner product, or equivalently the cosine squared of the angle between the lines the vectors  span, corresponds to the transition probability. Ray space, in mathematics known as projective Hilbert space, is the space of all unit vectors in Hilbert space up to the equivalence relation of differing by a phase factor. 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. As a consequence, the representation of a symmetry group on ray space can be lifted to a projective representation or sometimes even an ordinary representation on  Hilbert space. Rays and ray space. It is a postulate of quantum mechanics that vectors in Hilbert space that are scalar nonzero multiples of each other represent the same pure state. A ray belonging to the vector formula_1 is the complex line through the origin Two nonzero vectors formula_3 define the same ray, if and only if they differ by some nonzero complex number: formula_4, formula_5. Alternatively, we can consider a ray formula_6 as a set vectors with norm 1 that span the same line, a unit ray, by intersecting the line formula_6 with the unit sphere Two unit vectors  formula_3 then define the same unit ray formula_10 iff they differ by a phase factor: formula_11. This is the more usual picture in physics. The set of rays is in one to one correspondence with the set of unit rays and we can identify them. There is also a one-to-one correspondence between physical pure states formula_12 and (unit)rays formula_13 given by where formula_15 is the orthogonal projection on the line formula_13. In either interpretation, if formula_17 or formula_18 then formula_19 is a representative of formula_20. The space of all rays is called ray space. It can be defined  in several ways. One may define an equivalence relation formula_21 on formula_22 by Alternatively define a relation ≅ as an equivalence relation on the sphere formula_25. The unit ray space formula_26, is an incarnation of ray space defined (making no notational distinction with ray space) as the set of equivalence classes A third equivalent definition of ray space is as "pure state ray space" i.e. as density matrices that are orthogonal projections of rank 1 Each of these definitions make it clear that ray space is nothing but another name for projective Hilbert space. If formula_29 is finite, formula_30 has  real dimension formula_31. In fact, formula_26 is a compact complex manifold of dimension formula_33 which (by choosing a basis) is isomorphic to the projective space formula_34. For example, the Bloch sphere is isomorphic to the Riemann sphere formula_36. Ray space (i.e. projective space) takes a little getting used to, but is a very well studied object that predates quantum mechanics going back to the study of perspective by renaissance artists. It is not a vector space with well-defined linear combinations of rays. However, for every two vectors formula_3 and ratio of complex numbers formula_38 (i.e. element of formula_36) there  is a well defined ray formula_40. Moreover, for distinct rays formula_41 (i.e. linearly independent lines) there is a projective "line" of rays of the form formula_40 in formula_43: all 1 dimensional complex lines in the 2 complex dimensional plane spanned by formula_44 and formula_45 in formula_46). The Hilbert space structure on formula_46 defines additional structure on ray space. Define the ray correlation (or ray product) where formula_49 is the Hilbert space inner product, and formula_50 are representatives of formula_13 and formula_20. 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" states formula_53 and formula_19 in Hilbert space is given by i.e. we can define Born's rule on ray space by. Geometrically, we can define an angle formula_57 with formula_58 between the lines formula_13 and formula_20 by formula_61. The angle then turns out to satisfy the triangle inequality and defines a metric structure on ray space which comes from a Riemannian metric, the Fubini-Study metric. Symmetry transformations. Loosely speaking, a symmetry transformation is a change in which "nothing happens" or a "change in  our point of view" that does not change the outcomes of possible experiments. For example, translating a system in a homogeneous environment should have no qualitative effect on the outcomes of experiments made on the system. Likewise for rotating a system in an isotropic environment. This becomes even clearer when one considers the mathematically equivalent passive transformations, i.e. simply changes of coordinates and let the system be. Usually, the domain and range Hilbert spaces are the same. An exception would be (in a non-relativistic theory) the Hilbert space of electron states that is subjected to a charge conjugation transformation . In this case the electron states are mapped to the Hilbert space of positron states and vice versa. However this means that the symmetry acts on the direct sum of the Hilbert spaces. A transformation of a physical system is a transformation of states, hence mathematically a transformation, not of the Hilbert space, but of its ray space. Hence, in quantum mechanics, a transformation of a physical system gives rise to a bijective ray transformation formula_62 Since the composition of two physical transformations and the reversal of a physical transformation are also physical transformations, the set of all ray transformations  so obtained is a group acting on formula_30. Not all bijections of formula_30 are permissible as symmetry transformations, however. Physical transformations must preserve Born's rule. For a physical transformation, the transition probabilities in the transformed and untransformed systems should be preserved: A bijective ray transformation formula_67 is called a symmetry transformation iff A geometric interpretation, is that a symmetry transformation is an isometry of ray space. Some facts about symmetry transformations that can be verified using the definition: The set of symmetry transformations thus forms a group, the symmetry group of the system. Some important frequently occurring subgroups in  the symmetry group of a system are realizations of These groups are also referred to as symmetry groups of the system. Statement of Wigner's theorem. Preliminaries. Some preliminary definitions are needed to state the theorem. A transformation formula_69 of Hilbert spaces is unitary if it is bijective and formula_70 Since formula_71 for all formula_72, a unitary transformation is automatically linear and formula_73. Likewise, a transformation formula_74 is antiunitary if it is bijective and formula_75 As above, an antiunitary transformation is necessarily antilinear. Both variants are real linear and additive. Given a unitary transformation formula_76 of Hilbert spaces, define formula_77  This is a symmetry transformation since formula_78 In the same way an antiunitary transformation of Hilbert space induces a symmetry transformation. One says that a transformation formula_79 of Hilbert spaces is compatible with the transformation formula_80 of ray spaces if formula_81 or equivalently formula_82 for all formula_1. Transformations of Hilbert space induced by either a unitary linear transformation or an antiunitary antilinear operator are obviously compatible with the transformations or ray space they induce as described. Statement. Wigner's theorem states a converse of the above: Proofs can be found in , and . Antiunitary and antilinear transformations are less prominent in  physics. They are all related to a reversal of the direction of the flow of time. Remark 1: The significance of the uniqueness part of the theorem is that it specifies the degree of uniqueness of the representation on formula_46. For example, one might be tempted to believe that would be admissible, with formula_86 for formula_87 but this is not the case according to the theorem. In fact such a formula_88 would not be additive. Remark 2: Whether formula_62 must be represented by a unitary or antiunitary operator is determined by topology. If formula_90, the second cohomology formula_91 has a  unique generator formula_92 such that for a (equivalently for every) complex projective line formula_93, one has formula_94. Since formula_62 is a homeomorphism, formula_96 also generates formula_91 and so we have formula_98. If formula_76 is unitary, then formula_100 while if formula_74 is anti linear then formula_102. Representations and projective representations. If 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 with , then where the are ray transformations. From the uniqueness part of Wigner's theorem, one has for the compatible representatives , where is  a phase factor. The function is called a -cocycle or Schur multiplier. A map satisfying the above relation for some vector space is called a projective representation or a ray representation. If , then it is called a representation. One should note that the terminology differs between mathematics and physics. In the linked article, term "projective representation" has a slightly different meaning, but the term as presented here enters as an ingredient and the mathematics per se is of course the same. If the realization of the symmetry group, , is given in terms of action on the space of unit rays  , then it is a projective representation in the mathematical sense, while its representative on Hilbert space is a projective representation in the physical sense. Applying the last relation (several times) to the product and appealing to the known associativity of multiplication of operators on , one finds They also satisfy Upon redefinition of the phases, which is allowed by last theorem, one finds where the hatted quantities are defined by Utility of phase freedom. The following rather technical theorems and many more can be found, with accessible proofs, in . The freedom of choice of phases can be used to simplify the  phase factors. For some groups the phase can be eliminated altogether. In the case of the Lorentz group and its subgroup the rotation group SO(3), phases can, for projective representations, be chosen such that . For their respective universal covering groups, SL(2,C) and Spin(3), it is according to the theorem possible to have , i.e. they are proper representations. The study of redefinition of phases involves group cohomology. Two functions related as the hatted and non-hatted versions of above are said to be cohomologous. They belong to the same second cohomology class, i.e. they are represented by the same element in  , the second cohomology group of . If an element of contains the trivial function , then it is said to be trivial. The topic can be studied at the level of Lie algebras and Lie algebra cohomology as well. Assuming the projective representation 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.