Wigner's theorem, proved byEugene Wigner in 1931,^[2] is a cornerstone of themathematical formulation of quantum mechanics. The theorem specifies how physicalsymmetries such asrotations,translations, andCPT transformations are represented on theHilbert space ofstates.

The physical states in a quantum theory are represented byunit vectors in Hilbert space up to a phase factor, i.e. by the complex line orray the vector spans. In addition, by theBorn rule the absolute value of the unit vector'sinner product with a uniteigenvector, or equivalently thecosine squared of the angle between the lines the vectors span, corresponds to the transition probability.Ray space, in mathematics known asprojective 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 aunitary orantiunitary transformation of Hilbert space, which is unique up to a phase factor. As a consequence, the representation of asymmetry group on ray space can be lifted to aprojective representation or sometimes even an ordinaryrepresentation on Hilbert space.

It is apostulate of quantum mechanics that state vectors in complexseparableHilbert space${\displaystyle H}$  that are scalar nonzero multiples of each other represent the samepure state, i.e., the vectors${\displaystyle \mathrm{\Psi }\in H\setminus \{0\}}$  and${\displaystyle \lambda \mathrm{\Psi }}$ , with${\displaystyle \lambda \in \mathbb{C}\setminus \{0\}}$ , represent the same state.^[3] By multiplying the state vectors with thephase factor, one obtains a set of vectors called theray^[4][5]

${\displaystyle \underset{\_}{\mathrm{\Psi }}=\left\{e^{i\alpha }\mathrm{\Psi }:\alpha \in \mathbb{R}\right\}.}$ 

Two nonzero vectors${\displaystyle {\mathrm{\Psi }}_1,{\mathrm{\Psi }}_2}$  define the same ray, if and only if they differ by some nonzero complex number:${\displaystyle {\mathrm{\Psi }}_1=\lambda {\mathrm{\Psi }}_2}$ . Alternatively, we can consider a ray${\displaystyle \underset{\_}{\mathrm{\Psi }}}$  as a set of vectors with norm 1, aunit ray, by intersecting the line${\displaystyle \underset{\_}{\mathrm{\Psi }}}$  with the unit sphere^[6]

${\displaystyle SH=\{\mathrm{\Phi }\in H\mid \Vert \mathrm{\Phi }{\Vert }^2=1\}}$ .

Two unit vectors${\displaystyle {\mathrm{\Psi }}_1,{\mathrm{\Psi }}_2}$  then define the same unit ray${\displaystyle \underset{\_}{{\mathrm{\Psi }}_1}=\underset{\_}{{\mathrm{\Psi }}_2}}$  if they differ by a phase factor:${\displaystyle {\mathrm{\Psi }}_1=e^{i\alpha }{\mathrm{\Psi }}_2}$ . 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${\displaystyle \rho }$  and (unit) rays${\displaystyle \underset{\_}{\mathrm{\Phi }}}$  given by

${\displaystyle \rho =P_{\mathrm{\Phi }}=\frac{|\mathrm{\Phi }⟩⟨\mathrm{\Phi }|}{⟨\mathrm{\Phi }|\mathrm{\Phi }⟩}}$ 

where${\displaystyle P_{\mathrm{\Phi }}}$  is theorthogonal projection on the line${\displaystyle \underset{\_}{\mathrm{\Phi }}}$ . In either interpretation, if${\displaystyle \mathrm{\Phi }\in \underset{\_}{\mathrm{\Psi }}}$  or${\displaystyle P_{\mathrm{\Phi }}=P_{\mathrm{\Psi }}}$  then${\displaystyle \mathrm{\Phi }}$  is arepresentative of${\displaystyle \underset{\_}{\mathrm{\Psi }}}$ .^{[nb 1]}

The space of all rays is aprojective Hilbert space called theray space.^[7] It can be defined in several ways. One may define anequivalence relation${\displaystyle \sim }$  on${\displaystyle H\setminus \{0\}}$  by

${\displaystyle \mathrm{\Psi }\sim \mathrm{\Phi }\iff \mathrm{\Psi }=\lambda \mathrm{\Phi },\lambda \in \mathbb{C}\setminus \{0\},}$ 

and defineray space as thequotient set

${\displaystyle \mathbf{P}(H)=(H\setminus \{0\})/\sim }$ .

Alternatively, for an equivalence relation on the sphere${\displaystyle SH}$ , theunit ray space is an incarnation of ray space defined (making no notational distinction with ray space) as the set of equivalence classes

${\displaystyle \mathbf{P}(H)=SH/\sim }$ .

A third equivalent definition of ray space is aspure state ray space i.e. asdensity matrices that are orthogonal projections of rank 1^{[clarification needed]}

${\displaystyle \mathbf{P}(H)=\{P\in B(H)\mid P^2=P=P^{†},\mathbb{t}\mathbb{r}(P)=1\}}$ .

If${\displaystyle H}$  isn-dimensional, i.e.,${\displaystyle H_n:=H}$ , then${\displaystyle \mathbf{P}(H_n)}$  is isomorphic to thecomplex projective space${\displaystyle \mathbb{C}{\mathbf{P}}^{n-1}=\mathbf{P}({\mathbb{C}}^n)}$ . For example

${\displaystyle {\lambda }_1|+⟩+{\lambda }_2|-⟩,({\lambda }_1,{\lambda }_2)\in {\mathbb{C}}^2\setminus \{0\}}$ 

generate points on theBloch sphere; isomorphic to theRiemann sphere${\displaystyle \mathbb{C}{\mathbf{P}}^1}$ .

Ray space (i.e.projective space) isnot a vector space but rather a set ofvector lines (vector subspaces of dimension one) in a vector space of dimensionn + 1. For example, for every two vectors${\displaystyle {\mathrm{\Psi }}_1,{\mathrm{\Psi }}_2\in H_2}$  and ratio of complex numbers${\displaystyle ({\lambda }_1:{\lambda }_2)}$  (i.e. element of${\displaystyle \mathbb{C}{\mathbf{P}}^1}$ ) there is a well defined ray${\displaystyle \underset{\_}{{\lambda }_1{\mathrm{\Psi }}_1+{\lambda }_2{\mathrm{\Psi }}_2}}$ . As such, for distinct rays${\displaystyle {\underset{\_}{\mathrm{\Psi }}}_1,{\underset{\_}{\mathrm{\Psi }}}_2}$  (i.e. linearly independent lines) there is a projectiveline of rays of the form${\displaystyle \underset{\_}{{\lambda }_1{\mathrm{\Psi }}_1+{\lambda }_2{\mathrm{\Psi }}_2}}$  in${\displaystyle \mathbf{P}(H_2)}$ : all 1-dimensional complex lines in the 2-dimensional complex plane spanned by${\displaystyle {\mathrm{\Psi }}_1}$  and${\displaystyle {\mathrm{\Psi }}_2}$ . Contrarily to the case of vector spaces, however, an independent spanning set does not suffice for defining coordinates (see:projective frame).

The Hilbert space structure on${\displaystyle H}$  defines additional structure on ray space. Define theray correlation (orray product)

${\displaystyle \underset{\_}{\mathrm{\Psi }}\cdot \underset{\_}{\mathrm{\Phi }}=\frac{\left|⟨\mathrm{\Psi },\mathrm{\Phi }⟩\right|}{\Vert \mathrm{\Phi }\Vert \Vert \mathrm{\Psi }\Vert }=\sqrt{\mathrm{t}\mathrm{r}(P_{\mathrm{\Psi }}{P}_{\mathrm{\Phi }})},}$ 

where${\displaystyle ⟨,⟩}$  is the Hilbert spaceinner product, and${\displaystyle \mathrm{\Psi },\mathrm{\Phi }}$  are representatives of${\displaystyle \underset{\_}{\mathrm{\Phi }}}$  and${\displaystyle \underset{\_}{\mathrm{\Psi }}}$ . Note that the righthand side is independent of the choice of representatives.The physical significance of this definition is that according to theBorn rule, another postulate of quantum mechanics, thetransition probabilities betweennormalised states${\displaystyle \mathrm{\Psi }}$  and${\displaystyle \mathrm{\Phi }}$  in Hilbert space is given by

${\displaystyle P(\mathrm{\Psi }\to \mathrm{\Phi })=|⟨\mathrm{\Psi },\mathrm{\Phi }⟩{|}^2={\left(\underset{\_}{\mathrm{\Psi }}\cdot \underset{\_}{\mathrm{\Phi }}\right)}^2}$ 

i.e. we can define Born's rule on ray space by.

${\displaystyle P(\underset{\_}{\mathrm{\Psi }}\to \underset{\_}{\mathrm{\Phi }}):={\left(\underset{\_}{\mathrm{\Psi }}\cdot \underset{\_}{\mathrm{\Phi }}\right)}^2.}$ 

Geometrically, we can define an angle${\displaystyle \theta }$  with${\displaystyle 0\le \theta \le \pi /2}$  between the lines${\displaystyle \underset{\_}{\mathrm{\Phi }}}$  and${\displaystyle \underset{\_}{\mathrm{\Psi }}}$  by${\displaystyle \cos (\theta )=(\underset{\_}{\mathrm{\Psi }}\cdot \underset{\_}{\mathrm{\Phi }})}$ . The angle then turns out to satisfy the triangle inequality and defines a metric structure on ray space which comes from a Riemannian metric, theFubini-Study metric.

Loosely speaking, a symmetry transformation is a change in which "nothing happens"^[8] or a "change in our point of view"^[9] that does not change the outcomes of possible experiments. For example, translating a system in ahomogeneous environment should have no qualitative effect on the outcomes of experiments made on the system. Likewise for rotating a system in anisotropic environment. This becomes even clearer when one considers the mathematically equivalentpassive 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 acharge 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 abijectiveray transformation${\displaystyle T}$ 

 

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 agroup acting on${\displaystyle \mathbf{P}(H)}$ . Not all bijections of${\displaystyle \mathbf{P}(H)}$  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:

${\displaystyle P(\underset{\_}{\mathrm{\Psi }}\to \underset{\_}{\mathrm{\Phi }})={\left(\underset{\_}{\mathrm{\Psi }}\cdot \underset{\_}{\mathrm{\Phi }}\right)}^2={\left(T\underset{\_}{\mathrm{\Psi }}\cdot T\underset{\_}{\mathrm{\Phi }}\right)}^2=P\left(T\mathrm{\Psi }\to T\mathrm{\Phi }\right)}$ 

A bijective ray transformation${\displaystyle T:\mathbf{P}(H)\to \mathbf{P}(H)}$  is called asymmetry transformation iff^[10]:${\displaystyle T\underset{\_}{\mathrm{\Psi }}\cdot T\underset{\_}{\mathrm{\Phi }}=\underset{\_}{\mathrm{\Psi }}\cdot \underset{\_}{\mathrm{\Phi }},\mathrm{\forall }\underset{\_}{\mathrm{\Psi }},\underset{\_}{\mathrm{\Phi }}\in \mathbf{P}(H)}$ .A geometric interpretation is that a symmetry transformation is anisometry of ray space.

Some facts about symmetry transformations that can be verified using the definition:

• The product of two symmetry transformations, i.e. two symmetry transformations applied in succession, is a symmetry transformation.
• Any symmetry transformation has an inverse.
• The identity transformation is a symmetry transformation.
• Multiplication of symmetry transformations is associative.

The set of symmetry transformations thus forms agroup, thesymmetry group of the system. Some important frequently occurringsubgroups in the symmetry group of a system arerealizations of

• Thesymmetric group with its subgroups. This is important on the exchange of particle labels.
• ThePoincaré group. It encodes the fundamentalisometries ofspacetime – space-time translations andLorentz transformations
• Internal symmetry groups likeSU(2) andSU(3). They describe so calledinternal symmetries, likeisospin andcolor charge peculiar to quantum mechanical systems.

These groups are also referred to as symmetry groups of the system.

Some preliminary definitions are needed to state the theorem. A transformation${\displaystyle U:H\to K}$  between Hilbert spaces isunitary if it is bijective and

${\displaystyle ⟨U\mathrm{\Psi },U\mathrm{\Phi }⟩=⟨\mathrm{\Psi },\mathrm{\Phi }⟩}$ 

for all${\displaystyle \mathrm{\Psi },\mathrm{\Phi }}$  in${\displaystyle H}$ .If${\displaystyle H=K}$  then${\displaystyle U}$  reduces to aunitary operator whose inverse is equal to itsadjoint${\displaystyle U^{-1}=U^{†}}$ .

Likewise, a transformation${\displaystyle A:H\to K}$  isantiunitary if it is bijective and

${\displaystyle ⟨A\mathrm{\Psi },A\mathrm{\Phi }⟩=⟨\mathrm{\Psi },\mathrm{\Phi }{⟩}^{\ast }=⟨\mathrm{\Phi },\mathrm{\Psi }⟩.}$ 

Given a unitary transformation${\displaystyle U:H\to K}$  between Hilbert spaces, define

 

This is a symmetry transformation since$${\displaystyle T_U\underset{\_}{\mathrm{\Psi }}\cdot T_U\underset{\_}{\mathrm{\Phi }}=\frac{\left|⟨U\mathrm{\Psi },U\mathrm{\Phi }⟩\right|}{\Vert U\mathrm{\Psi }\Vert \Vert U\mathrm{\Phi }\Vert }=\frac{\left|⟨\mathrm{\Psi },\mathrm{\Phi }⟩\right|}{\Vert \mathrm{\Psi }\Vert \Vert \mathrm{\Phi }\Vert }=\underset{\_}{\mathrm{\Psi }}\cdot \underset{\_}{\mathrm{\Phi }}.}$$ 

In the same way an antiunitary transformation between Hilbert space induces a symmetry transformation. One says that a transformation${\displaystyle U:H\to K}$  between Hilbert spaces iscompatible with the transformation${\displaystyle T:\mathbf{P}(H)\to \mathbf{P}(K)}$  between ray spaces if${\displaystyle T=T_U}$  or equivalently

${\displaystyle U\mathrm{\Psi }\in T\underset{\_}{\mathrm{\Psi }}}$ 

for all${\displaystyle \mathrm{\Psi }\in H\setminus \{0\}}$ .^[11]

Wigner's theorem states a converse of the above:^[12]

Proofs can be found in Wigner (1931,1959),Uhlhorn (1963),Bargmann (1964) andWeinberg (2002).Antiunitary transformations are less prominent in physics. They are all related to a reversal of the direction of the flow of time.^[13]

Remark 1: The significance of the uniqueness part of the theorem is that it specifies the degree of uniqueness of the representation on${\displaystyle H}$ . For example, one might be tempted to believe that

${\displaystyle V\mathrm{\Psi }=U{e}^{i\alpha (\mathrm{\Psi })}\mathrm{\Psi },\alpha (\mathrm{\Psi })\in \mathbb{R},\mathrm{\Psi }\in H(\text{wrong unless }\alpha (\mathrm{\Psi })\text{ is const.})}$ 

would be admissible, with${\displaystyle \alpha (\mathrm{\Psi })\ne \alpha (\mathrm{\Phi })}$  for${\displaystyle ⟨\mathrm{\Psi },\mathrm{\Phi }⟩=0}$  but this is not the case according to the theorem.^{[nb 2][14]} In fact such a${\displaystyle V}$  would not be additive.

Remark 2: Whether${\displaystyle T}$  must be represented by a unitary or antiunitary operator is determined by topology. If${\displaystyle {\dim }_{\mathbb{C}}(\mathbb{P}H)={\dim }_{\mathbb{C}}(\mathbb{P}K)\ge 1}$ , the secondcohomology${\displaystyle H^2(\mathbb{P}H)}$  has a unique generator${\displaystyle c_{\mathbb{P}H}}$  such that for a (equivalently for every) complex projective line${\displaystyle L\subset \mathbb{P}H}$ , one has${\displaystyle c_{\mathbb{P}H}\cap [L]={\mathrm{deg}}_L(c_{\mathbb{P}H}{|}_L)=1}$ . Since${\displaystyle T}$  is a homeomorphism,${\displaystyle T^{\ast }{c}_{\mathbb{P}K}}$  also generates${\displaystyle H^2(\mathbb{P}H)}$  and so we have${\displaystyle T^{\ast }{c}_{\mathbb{P}K}=\pm c_{\mathbb{P}H}}$ . If${\displaystyle U:H\to K}$  is unitary, then${\displaystyle T_U^{\ast }{c}_{\mathbb{P}K}=c_{\mathbb{P}H}}$  while if${\displaystyle A:H\to K}$  is anti linear then${\displaystyle T_A^{\ast }{c}_{\mathbb{P}K}=-c_{\mathbb{P}H}}$ .

Remark 3: Wigner's theorem is in close connection with thefundamental theorem of projective geometry^[15],^[16]

IfG 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 iff,g,h ∈G withfg =h, then

${\displaystyle T(f)T(g)=T(h),}$ 

where theT are ray transformations. From the uniqueness part of Wigner's theorem, one has for the compatible representativesU,

${\displaystyle U(f)U(g)=\omega (f,g)U(fg)=e^{i\xi (f,g)}U(fg),}$ 

whereω(f,g) is a phase factor.^{[nb 3]}

The functionω is called a2-cocycle orSchur multiplier. A mapU:G → GL(V) satisfying the above relation for some vector spaceV is called aprojective representation or aray representation. Ifω(f,g) = 1, then it is called arepresentation.

One should note that the terminology differs between mathematics and physics. In the linked article, termprojective 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,g →T(g), is given in terms of action on the space of unit raysS =PH, then it is a projective representationG → PGL(H) in the mathematical sense, while its representative on Hilbert space is a projective representationG → GL(H) in the physical sense.

Applying the last relation (several times) to the productfgh and appealing to the known associativity of multiplication of operators onH, one finds

 

They also satisfy

 

Upon redefinition of the phases,

${\displaystyle U(g)\mapsto \hat{U}(g)=\eta (g)U(g)=e^{i\zeta (g)}U(g),}$ 

which is allowed by last theorem, one finds^[17][18]

 

where the hatted quantities are defined by

${\displaystyle \hat{U}(f)\hat{U}(g)=\hat{\omega }(f,g)\hat{U}(fg)=e^{i\hat{\xi }(f,g)}\hat{U}(fg).}$ 

The following rather technical theorems and many more can be found, with accessible proofs, inBargmann (1954).

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 theLorentz group and its subgroup therotation group SO(3), phases can, for projective representations, be chosen such thatω(g,h) = ± 1. For their respectiveuniversal covering groups,SL(2,C) andSpin(3), it is according to the theorem possible to haveω(g,h) = 1, i.e. they are proper representations.

The study of redefinition of phases involvesgroup cohomology. Two functions related as the hatted and non-hatted versions ofω above are said to becohomologous. They belong to the samesecond cohomology class, i.e. they are represented by the same element inH²(G), thesecond cohomology group ofG. If an element ofH²(G) contains the trivial functionω = 0, then it is said to betrivial.^[18] The topic can be studied at the level ofLie algebras andLie algebra cohomology as well.^[20][21]

Assuming the projective representationg →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]}

Wigner's theorem applies toautomorphisms on the Hilbert space of pure states. Theorems by Kadison^[23] and Simon^[24] apply to the space of mixed states (trace-class positive operators) and use slight different notions of symmetry.^[25][26]

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