Wigner's theorem, proved by Eugene Wigner in 1931,[1] is a cornerstone of the mathematical formulation of quantum mechanics. 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. 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.
H
\Psi\inH\setminus\{0\}
λ\Psi
λ\inC\setminus\{0\}
\underline{\Psi}=\left\{ei\alpha\Psi:\alpha\inR\right\}.
\Psi1,\Psi2
\Psi1=λ\Psi2
\underline\Psi
\underline\Psi
SH=\{\Phi\inH\mid\|\Phi\|2=1\}
\Psi1,\Psi2
\underline{\Psi1}=\underline{\Psi2}
\Psi1=ei\alpha\Psi2
\rho
\underline{\Phi}
\rho=P\Phi=
|\Phi\rangle\langle\Phi| | |
\langle\Phi|\Phi\rangle |
P\Phi
\underline{\Phi}
\Phi\in\underline{\Psi}
P\Phi=P\Psi
\Phi
\underline{\Psi}
\sim
H\setminus\{0\}
\Psi\sim\Phi\Leftrightarrow\Psi=λ\Phi, λ\inC\setminus\{0\},
P(H)=(H\setminus\{0\})/{\sim}
Alternatively, for an equivalence relation on the sphere
SH
P(H)=SH/\sim
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
P(H)=\{P\inB(H)\midP2=P=P\dagger,tr(P)=1\}
H
Hn:=H
P(Hn)
CPn-1=P(Cn)
λ1|+\rangle+λ2|-\rangle, (λ1,λ2)\inC2\setminus\{0\}
CP1
\Psi1,\Psi2\inH2
(λ1:λ2)
CP1
\underline{λ1\Psi1+λ2\Psi2}
\underline{\Psi}1,\underline{\Psi}2
\underline{λ1\Psi1+λ2\Psi2}
P(H2)
\Psi1
\Psi2
The Hilbert space structure on
H
\underline{\Psi} ⋅ \underline{\Phi}=
\left|\left\langle\Psi,\Phi\right\rangle\right| | |
\|\Phi\|\|\Psi\| |
=\sqrt{tr(P\PsiP\Phi)},
\langle,\rangle
\Psi,\Phi
\underline{\Phi}
\underline{\Psi}
\Psi
\Phi
P(\Psi → \Phi)=|\langle\Psi,\Phi\rangle|2=\left(\underline{\Psi} ⋅ \underline{\Phi}\right)2
P(\underline{\Psi}\to\underline{\Phi}):=\left(\underline{\Psi} ⋅ \underline{\Phi}\right)2.
\theta
0\le\theta\le\pi/2
\underline{\Phi}
\underline{\Psi}
\cos(\theta)=(\underline{\Psi} ⋅ \underline{\Phi})
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
T
\begin{align} T:P(H)&\toP(H)\\ \underline{\Psi}&\mapstoT\underline{\Psi}.\\ \end{align}
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
P(H)
P(H)
For a physical transformation, the transition probabilities in the transformed and untransformed systems should be preserved:
P(\underline{\Psi} → \underline{\Phi})=\left(\underline{\Psi} ⋅ \underline{\Phi}\right)2=\left(T\underline{\Psi} ⋅ T\underline{\Phi}\right)2=P\left(T\Psi → T\Phi\right)
P(H)\toP(H)
T\underline{\Psi} ⋅ T\underline{\Phi}=\underline{\Psi} ⋅ \underline{\Phi}, \forall\underline\Psi,\underline\Phi\inP(H)
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.
Some preliminary definitions are needed to state the theorem. A transformation
U:H\toK
\langleU\Psi,U\Phi\rangle=\langle\Psi,\Phi\rangle.
H=K
U
U-1=U\dagger
Likewise, a transformation
A:H\toK
\langleA\Psi,A\Phi\rangle=\langle\Psi,\Phi\rangle*=\langle\Phi,\Psi\rangle.
U:H\toK
\begin{align} TU:P(H)&\toP(K)\\ \underline{\Psi}&\mapsto\underline{U\Psi}\\ \end{align}
This is a symmetry transformation since
In the same way an antiunitary transformation between Hilbert space induces a symmetry transformation. One says that a transformation
U:H\toK
T:P(H)\toP(K)
T=TU
U\Psi\inT\underline\Psi
\Psi\inH\setminus\{0\}
Wigner's theorem states a converse of the above: Proofs can be found in, and .Antiunitary 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
H
V\Psi=Uei\alpha(\Psi)\Psi,\alpha(\Psi)\inR,\Psi\inH (wrongunless\alpha(\Psi)isconst.)
\alpha(\Psi)\ne\alpha(\Phi)
\langle\Psi,\Phi\rangle=0
V
Remark 2: Whether
T
\dimC(PH)=\dimC(PK)\ge1
H2(PH)
cPH
L\subsetPH
cPH\cap[L]=\degL(cPH|L)=1
T
*c | |
T | |
PK |
H2(PH)
*c | |
T | |
PK |
=\pmcPH
U:H\toK
*c | |
T | |
PK |
=cPH
A:H\toK
*c | |
T | |
PK |
=-cPH
Remark 3: Wigner's theorem is in close connection with the fundamental theorem of projective geometry
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
T(f)T(g)=T(h),
where the are ray transformations. From the uniqueness part of Wigner's theorem, one has for the compatible representatives,
U(f)U(g)=\omega(f,g)U(fg)=ei\xi(f,U(fg),
where is a phase factor.[5]
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
\begin{align} \omega(f,g)\omega(fg,h)&=\omega(g,h)\omega(f,gh),\\ \xi(f,g)+\xi(fg,h)&=\xi(g,h)+\xi(f,gh) (\operatorname{mod}2\pi). \end{align}
They also satisfy
\begin{align} \omega(g,e)&=\omega(e,g)=1,\\ \xi(g,e)&=\xi(e,g)=0 (\operatorname{mod}2\pi),\\ \omega\left(g,g-1\right)&=\omega(g-1,g),\\ \xi\left(g,g-1\right)&=\xi(g-1,g).\\ \end{align}
Upon redefinition of the phases,
U(g)\mapsto\hat{U}(g)=η(g)U(g)=ei\zeta(g)U(g),
which is allowed by last theorem, one finds[6]
\begin{align} \hat{\omega}(g,h)&=\omega(g,h)η(g)η(h)η(gh)-1,\\ \hat{\xi}(g,h)&=\xi(g,h)+\zeta(g)+\zeta(h)-\zeta(gh) (\operatorname{mod}2\pi),\end{align}
where the hatted quantities are defined by
\hat{U}(f)\hat{U}(g)=\hat{\omega}(f,g)\hat{U}(fg)=ei\hat{\xi(f,g)}\hat{U}(fg).
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.[7]
Wigner's theorem applies to automorphisms on the Hilbert space of pure states. Theorems by Kadison[8] and Simon[9] apply to the space of mixed states (trace-class positive operators) and use slight different notions of symmetry.[10] [11]
H
\Psi