Symmetries in quantum mechanics describe features of spacetime and particles which are unchanged under some transformation, in the context of quantum mechanics, relativistic quantum mechanics and quantum field theory, and with applications in the mathematical formulation of the standard model and condensed matter physics. In general, symmetry in physics, invariance, and conservation laws, are fundamentally important constraints for formulating physical theories and models. In practice, they are powerful methods for solving problems and predicting what can happen. While conservation laws do not always give the answer to the problem directly, they form the correct constraints and the first steps to solving a multitude of problems. In application, understanding symmetries can also provide insights on the eigenstates that can be expected. For example, the existence of degenerate states can be inferred by the presence of non-commuting symmetry operators or that the non-degenerate states are also eigenvectors of symmetry operators. This article outlines the connection between the classical form of continuous symmetries as well as their quantum operators, and relates them to the Lie groups, and relativistic transformations in the Lorentz group and Poincaré group.
Notation The notational conventions used in this article are as follows. Boldface indicates vectors, four vectors, matrices, and vectorial operators, while quantum states use bra–ket notation. Wide hats are for operators, narrow hats are for unit vectors (including their components in tensor index notation). The summation convention on the repeated tensor indices is used, unless stated otherwise. The Minkowski metric signature is (+−−−).
Symmetry transformations on the wavefunction in non-relativistic quantum mechanics
Continuous symmetries Generally, the correspondence between continuous symmetries and conservation laws is given by Noether's theorem. The form of the fundamental quantum operators, for example the energy operator as a partial time derivative and momentum operator as a spatial gradient, becomes clear when one considers the initial state, then changes one parameter of it slightly. This can be done for displacements (lengths), durations (time), and angles (rotations). Additionally, the invariance of certain quantities can be seen by making such changes in lengths and angles, illustrating conservation of these quantities. In what follows, transformations on only one-particle wavefunctions in the form:
Ω ^ ψ ( r , t ) = ψ ( r ′ , t ′ ) {\displaystyle {\widehat {\Omega }}\psi (\mathbf {r} ,t)=\psi (\mathbf {r} ',t')}
are considered, where Ω ^ {\displaystyle {\widehat {\Omega }}} denotes a unitary operator. Unitarity is generally required for operators representing transformations of space, time, and spin, since the norm of a state (representing the total probability of finding the particle somewhere with some spin) must be invariant under these transformations. The inverse of a unitary operator is its Hermitian conjugate Ω ^ − 1 = Ω ^ † {\displaystyle {\widehat {\Omega }}^{-1}={\widehat {\Omega }}^{\dagger }} . The results can be extended to many-particle wavefunctions. Written in Dirac notation as standard, the transformations on quantum state vectors are:
Ω ^ | r ( t ) ⟩ = | r ′ ( t ′ ) ⟩ {\displaystyle {\widehat {\Omega }}\left|\mathbf {r} (t)\right\rangle =\left|\mathbf {r} '(t')\right\rangle }
Now, the action of Ω ^ {\displaystyle {\widehat {\Omega }}} changes ψ(r, t) to ψ(r′, t′), so the inverse Ω ^ − 1 = Ω ^ † {\displaystyle {\widehat {\Omega }}^{-1}={\widehat {\Omega }}^{\dagger }} changes ψ(r′, t′) back to ψ(r, t). Thus, an operator A ^ {\displaystyle {\widehat {A}}} invariant under Ω ^ {\displaystyle {\widehat {\Omega }}} satisfies:
A ^ ψ = Ω ^ † A ^ Ω ^ ψ ⇒ Ω ^ A ^ ψ = A ^ Ω ^ ψ . {\displaystyle {\widehat {A}}\psi ={\widehat {\Omega }}^{\dagger }{\widehat {A}}{\widehat {\Omega }}\psi \quad \Rightarrow \quad {\widehat {\Omega }}{\widehat {A}}\psi ={\widehat {A}}{\widehat {\Omega }}\psi .}
Concomitantly,
[ Ω ^ , A ^ ] ψ = 0 {\displaystyle [{\widehat {\Omega }},{\widehat {A}}]\psi =0}
for any state ψ (i.e. Ω ^ {\displaystyle {\widehat {\Omega }}} and A ^ {\displaystyle {\widehat {A}}} commute). Quantum operators representing observables are also required to be Hermitian so that their eigenvalues are real numbers, i.e. the operator equals its Hermitian conjugate, A ^ = A ^ † {\displaystyle {\widehat {A}}={\widehat {A}}^{\dagger }} .
Overview of Lie group theory
Following are the key points of group theory relevant to quantum theory, examples are given throughout the article. For an alternative approach using matrix groups, see the books of Hall Let G be a Lie group, which is a group that locally is parameterized by a finite number N of real continuously varying parameters ξ1, ξ2, ..., ξN. In more mathematical language, this means that G is a smooth manifold that is also a group, for which the group operations are smooth.
the dimension of the group, N, is the number of parameters it has. the group elements, g, in G are functions of the parameters: g = G ( ξ 1 , ξ 2 , … ) {\displaystyle g=G(\xi _{1},\xi _{2},\dots )} and all parameters set to zero returns the identity element of the group: I = G ( 0 , 0 , … ) {\displaystyle I=G(0,0,\dots )} Group elements are often matrices which act on vectors, or transformations acting on functions. The generators of the group are the partial derivatives of the group elements with respect to the group parameters with the result evaluated when the parameter is set to zero: X j = ∂ g ∂ ξ j | ξ j = 0 {\displaystyle X_{j}=\left.{\frac {\partial g}{\partial \xi _{j}}}\right|_{\xi _{j}=0}} In the language of manifolds, the generators are the elements of the tangent space to G at the identity. The generators are also known as infinitesimal group elements or as the elements of the Lie algebra of G. (See the discussion below of the commutator.) One aspect of generators in theoretical physics is they can be constructed themselves as operators corresponding to symmetries, which may be written as matrices, or as differential operators. In quantum theory, for unitary representations of the group, the generators require a factor of i: X j = i ∂ g ∂ ξ j | ξ j = 0 {\displaystyle X_{j}=i\left.{\frac {\partial g}{\partial \xi _{j}}}\right|_{\xi _{j}=0}} The generators of the group form a vector space, which means linear combinations of generators also form a generator. The generators (whether matrices or differential operators) satisfy the commutation relations: [ X a , X b ] = i f a b c X c {\displaystyle \left[X_{a},X_{b}\right]=if_{abc}X_{c}} where fabc are the (basis dependent) structure constants of the group. This makes, together with the vector space property, the set of all generators of a group a Lie algebra. Due to the antisymmetry of the bracket, the structure constants of the group are antisymmetric in the first two indices. The representations of the group then describe the ways that the group G (or its Lie algebra) can act on a vector space. (The vector space might be, for example, the space of eigenvectors for a Hamiltonian having G as its symmetry group.) We denote the representations using a capital D. One can then differentiate D to obtain a representation of the Lie algebra, often also denoted by D. These two representations are related as follows: D [ g ( ξ j ) ] ≡ D ( ξ j ) = e i ξ j D ( X j ) {\displaystyle D[g(\xi _{j})]\equiv D(\xi _{j})=e^{i\xi _{j}D(X_{j})}} without summation on the repeated index j. Representations are linear operators that take in group elements and preserve the composition rule: D ( ξ a ) D ( ξ b ) = D ( ξ a ξ b ) . {\displaystyle D(\xi _{a})D(\xi _{b})=D(\xi _{a}\xi _{b}).}
A representation which cannot be decomposed into a direct sum of other representations, is called irreducible. It is conventional to label irreducible representations by a superscripted number n in brackets, as in D(n), or if there is more than one number, we write D(n, m, ...). There is an additional subtlety that arises in quantum theory, where two vectors that differ by multiplication by a scalar represent the same physical state. Here, the pertinent notion of representation is a projective representation, one that only satisfies the composition law up to a scalar. In the context of quantum mechanical spin, such representations are called spinorial.
Momentum and energy as generators of translation and time evolution, and rotation The space translation operator T ^ ( Δ r ) {\displaystyle {\widehat {T}}(\Delta \mathbf {r} )} acts on a wavefunction to shift the space coordinates by an infinitesimal displacement Δr. The explicit expression T ^ {\displaystyle {\widehat {T}}} can be quickly determined by a Taylor expansion of ψ(r + Δr, t) about r, then (keeping the first order term and neglecting second and higher order terms), replace the space derivatives by the momentum operator p ^ {\displaystyle {\widehat {\mathbf {p} }}} . Similarly for the time translation operator acting on the time parameter, the Taylor expansion of ψ(r, t + Δt) is about t, and the time derivative replaced by the energy operator E ^ {\displaystyle {\widehat {E}}} .
The exponential functions arise by definition as those limits, due to Euler, and can be understood physically and mathematically as follows. A net translation can be composed of many small translations, so to obtain the translation operator for a finite increment, replace Δr by Δr/N and Δt by Δt/N, where N is a positive non-zero integer. Then as N increases, the magnitude of Δr and Δt become even smaller, while leaving the directions unchanged. Acting the infinitesimal operators on the wavefunction N times and taking the limit as N tends to infinity gives the finite operators. Space and time translations commute, which means the operators and generators commute.
For a time-independent Hamiltonian, energy is conserved in time and quantum states are stationary states: the eigenstates of the Hamiltonian are the energy eigenvalues E:
U ^ ( t ) = exp ( − i Δ t E ℏ ) {\displaystyle {\widehat {U}}(t)=\exp \left(-{\frac {i\Delta tE}{\hbar }}\right)}
and all stationary states have the form
ψ ( r , t + t 0 ) = U ^ ( t − t 0 ) ψ ( r , t 0 ) {\displaystyle \psi (\mathbf {r} ,t+t_{0})={\widehat {U}}(t-t_{0})\psi (\mathbf {r} ,t_{0})}
where t0 is the initial time, usually set to zero since there is no loss of continuity when the initial time is set. An alternative notation is U ^ ( t − t 0 ) ≡ U ( t , t 0 ) {\displaystyle {\widehat {U}}(t-t_{0})\equiv U(t,t_{0})} .
Angular momentum as the generator of rotations
Orbital angular momentum The rotation operator, R ^ {\displaystyle {\widehat {R}}} , acts on a wavefunction to rotate the spatial coordinates of a particle by a constant angle Δθ:
R ^ ( Δ θ , a ^ ) ψ ( r , t ) = ψ ( r ′ , t ) {\displaystyle {\widehat {R}}(\Delta \theta ,{\hat {\mathbf {a} }})\psi (\mathbf {r} ,t)=\psi (\mathbf {r} ',t)}
where r′ are the rotated coordinates about an axis defined by a unit vector a ^ = ( a 1 , a 2 , a 3 ) {\displaystyle {\hat {\mathbf {a} }}=(a_{1},a_{2},a_{3})} through an angular increment Δθ, given by:
r ′ = R ^ ( Δ θ , a ^ ) r . {\displaystyle \mathbf {r} '={\widehat {R}}(\Delta \theta ,{\hat {\mathbf {a} }})\mathbf {r} \,.}
where R ^ ( Δ θ , a ^ ) {\displaystyle {\widehat {R}}(\Delta \theta ,{\hat {\mathbf {a} }})} is a rotation matrix dependent on the axis and angle. In group theoretic language, the rotation matrices are group elements, and the angles and axis Δ θ a ^ = Δ θ ( a 1 , a 2 , a 3 ) {\displaystyle \Delta \theta {\hat {\mathbf {a} }}=\Delta \theta (a_{1},a_{2},a_{3})} are the parameters, of the three-dimensional special orthogonal group, SO(3). The rotation matrices about the standard Cartesian basis vector e ^ x , e ^ y , e ^ z {\displaystyle {\hat {\mathbf {e} }}_{x},{\hat {\mathbf {e} }}_{y},{\hat {\mathbf {e} }}_{z}} through angle Δθ, and the corresponding generators of rotations J = (Jx, Jy, Jz), are:
More generally for rotations about an axis defined by a ^ {\displaystyle {\hat {\mathbf {a} }}} , the rotation matrix elements are:
[ R ^ ( θ , a ^ ) ] i j = ( δ i j − a i a j ) cos θ − ε i j k a k sin θ + a i a j {\displaystyle [{\widehat {R}}(\theta ,{\hat {\mathbf {a} }})]_{ij}=(\delta _{ij}-a_{i}a_{j})\cos \theta -\varepsilon _{ijk}a_{k}\sin \theta +a_{i}a_{j}}
where δij is the Kronecker delta, and εijk is the Levi-Civita symbol. It is not as obvious how to determine the rotational operator compared to space and time translations. We may consider a special case (rotations about the x, y, or z-axis) then infer the general result, or use the general rotation matrix directly and tensor index notation with δij and εijk. To derive the infinitesimal rotation operator, which corresponds to small Δθ, we use the small angle approximations sin(Δθ) ≈ Δθ and cos(Δθ) ≈ 1, then Taylor expand about r or ri, keep the first order term, and substitute the angular momentum operator components.
The z-component of angular momentum can be replaced by the component along the axis defined by a ^ {\displaystyle {\hat {\mathbf {a} }}} , using the dot product a ^ ⋅ L ^ {\displaystyle {\hat {\mathbf {a} }}\cdot {\widehat {\mathbf {L} }}} . Again, a finite rotation can be made from many small rotations, replacing Δθ by Δθ/N and taking the limit as N tends to infinity gives the rotation operator for a finite rotation. Rotations about the same axis do commute, for example a rotation through angles θ1 and θ2 about axis i can be written
R ( θ 1 + θ 2 , e i ) = R ( θ 1 e i ) R ( θ 2 e i ) , [ R ( θ 1 e i ) , R ( θ 2 e i ) ] = 0 . {\displaystyle R(\theta _{1}+\theta _{2},\mathbf {e} _{i})=R(\theta _{1}\mathbf {e} _{i})R(\theta _{2}\mathbf {e} _{i})\,,\quad [R(\theta _{1}\mathbf {e} _{i}),R(\theta _{2}\mathbf {e} _{i})]=0\,.}
However, rotations about different axes do not commute. The general commutation rules are summarized by
[ L i , L j ] = i ℏ ε i j k L k . {\displaystyle [L_{i},L_{j}]=i\hbar \varepsilon _{ijk}L_{k}.}
In this sense, orbital angular momentum has the common sense properties of rotations. Each of the above commutators can be easily demonstrated by holding an everyday object and rotating it through the same angle about any two different axes in both possible orderings; the final configurations are different. In quantum mechanics, there is another form of rotation which mathematically appears similar to the orbital case, but has different properties, described next.
Spin angular momentum All previous quantities have classical definitions. Spin is a quantity possessed by particles in quantum mechanics without any classical analogue, having the units of angular momentum. The spin vector operator is denoted S ^ = ( S x ^ , S y ^ , S z ^ ) {\displaystyle {\widehat {\mathbf {S} }}=({\widehat {S_{x}}},{\widehat {S_{y}}},{\widehat {S_{z}}})} . The eigenvalues of its components are the possible outcomes (in units of ℏ {\displaystyle \hbar } ) of a measurement of the spin projected onto one of the basis directions. Rotations (of ordinary space) about an axis a ^ {\displaystyle {\hat {\mathbf {a} }}} through angle θ about the unit vector a ^ {\displaystyle {\hat {a}}} in space acting on a multicomponent wave function (spinor) at a point in space is represented by:
However, unlike orbital angular momentum in which the z-projection quantum number ℓ can only take positive or negative integer values (including zero), the z-projection spin quantum number s can take all positive and negative half-integer values. There are rotational matrices for each spin quantum number. Evaluating the exponential for a given z-projection spin quantum number s gives a (2s + 1)-dimensional spin matrix. This can be used to define a spinor as a column vector of 2s + 1 components which transforms to a rotated coordinate system according to the spin matrix at a fixed point in space. For the simplest non-trivial case of s = 1/2, the spin operator is given by
S ^ = ℏ 2 σ {\displaystyle {\widehat {\mathbf {S} }}={\frac {\hbar }{2}}{\boldsymbol {\sigma }}}
where the Pauli matrices in the standard representation are:
σ 1 = σ x = ( 0 1 1 0 ) , σ 2 = σ y = ( 0 − i i 0 ) , σ 3 = σ z = ( 1 0 0 − 1 ) {\displaystyle \sigma _{1}=\sigma _{x}={\begin{pmatrix}0&1\\1&0\end{pmatrix}}\,,\quad \sigma _{2}=\sigma _{y}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}}\,,\quad \sigma _{3}=\sigma _{z}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}}
Total angular momentum The total angular momentum operator is the sum of the orbital and spin
J ^ = L ^ + S ^ {\displaystyle {\widehat {\mathbf {J} }}={\widehat {\mathbf {L} }}+{\widehat {\mathbf {S} }}}
and is an important quantity for multi-particle systems, especially in nuclear physics and the quantum chemistry of multi-electron atoms and molecules. We have a similar rotation matrix:
J ^ ( θ , a ^ ) = exp ( − i ℏ θ a ^ ⋅ J ^ ) {\displaystyle {\widehat {J}}(\theta ,{\hat {\mathbf {a} }})=\exp \left(-{\frac {i}{\hbar }}\theta {\hat {\mathbf {a} }}\cdot {\widehat {\mathbf {J} }}\right)}
Conserved quantities in the quantum harmonic oscillator The dynamical symmetry group of the n dimensional quantum harmonic oscillator is the special unitary group SU(n). As an example, the number of infinitesimal generators of the corresponding Lie algebras of SU(2) and SU(3) are three and eight respectively. This leads to exactly three and eight independent conserved quantities (other than the Hamiltonian) in these systems. The two dimensional quantum harmonic oscillator has the expected conserved quantities of the Hamiltonian and the angular momentum, but has additional hidden conserved quantities of energy level difference and another form of angular momentum.
Lorentz group in relativistic quantum mechanics Following is an overview of the Lorentz group; a treatment of boosts and rotations in spacetime. Throughout this section, see (for example) T. Ohlsson (2011) and E. Abers (2004). Lorentz transformations can be parametrized by rapidity φ for a boost in the direction of a three-dimensional unit vector n ^ = ( n 1 , n 2 , n 3 ) {\displaystyle {\hat {\mathbf {n} }}=(n_{1},n_{2},n_{3})} , and a rotation angle θ about a three-dimensional unit vector a ^ = ( a 1 , a 2 , a 3 ) {\displaystyle {\hat {\mathbf {a} }}=(a_{1},a_{2},a_{3})} defining an axis, so φ n ^ = φ ( n 1 , n 2 , n 3 ) {\displaystyle \varphi {\hat {\mathbf {n} }}=\varphi (n_{1},n_{2},n_{3})} and θ a ^ = θ ( a 1 , a 2 , a 3 ) {\displaystyle \theta {\hat {\mathbf {a} }}=\theta (a_{1},a_{2},a_{3})} are together six parameters of the Lorentz group (three for rotations and three for boosts). The Lorentz group is 6-dimensional.
Pure rotations in spacetime The rotation matrices and rotation generators considered above form the spacelike part of a four-dimensional matrix, representing pure-rotation Lorentz transformations. Three of the Lorentz group elements R ^ x , R ^ y , R ^ z {\displaystyle {\widehat {R}}_{x},{\widehat {R}}_{y},{\widehat {R}}_{z}} and generators J = (J1, J2, J3) for pure rotations are:
The rotation matrices act on any four vector A = (A0, A1, A2, A3) and rotate the space-like components according to
A ′ =