3.1. Wigner’s Definition of Symmetry

In this section, I account for the manner in which symmetries and conservation laws relate to each other in quantum mechanics. In order to do this, I will follow Wigner’s own writings on the subject. My discussion will be based, in particular, on four journal articles that Wigner published in 1927, on an additional paper that he published the year after that, and on his 1931 classic Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra (Wigner Reference Wigner1927a, Reference Wigner1927b, Reference Wigner1927c, Reference Wigner1927d, Reference Wigner1928, Reference Wigner and Griffin1959).

Wigner’s first step in his work of the late 1920s and early 1930s was to provide a notion of symmetry that was appropriate for the context of quantum mechanics. Wigner drew from the familiar conception of symmetries as transformations of the dynamical variables of a theory that leave the explicit form of its equations of motion unchanged. This requirement, as is well known, also acts as a necessary and sufficient condition for a group of transformations to map the set of solutions of the equations of motion of a theory onto itself. Owing to this, the symmetries of the equations of motion of a theory are sometimes known as “dynamical” symmetries (Brown and Holland Reference Brown and Holland2004).

In order to make use of the notion of a dynamical symmetry in his work on quantum mechanics, Wigner had to refine this concept in the following ways. First, he had to specify the symmetries of what equation he was going to study. Wigner invariably turned to the time-independent Schrödinger equation

(1)$\begin{array}{}\hat{H}{\psi }_E=E{\psi }_E,\end{array}$

which tells us that the eigenfunctions ψ_E of the Hamiltonian operator Ĥ are energy eigenfunctions.

Second, Wigner had to clarify what was the space that symmetry transformations were going to act on, in the case of quantum mechanics. Wigner’s writings typically start with him considering transformations R of the position coordinates on which the wave function depends. In quantum mechanics, however, it is much more natural to think in terms of operators acting on the wave function itself. In order to be able to formulate his definition of symmetry in these terms, Wigner showed that for any transformation R ⁻¹ that acts on the arguments of the wave function we can always define an operator ${\hat{P}}_R$,

(2)$\begin{array}{}{\hat{P}}_R\psi \left(\overrightarrow{r},t\right)=\psi \left(R^{-1}\overrightarrow{r},t\right),\end{array}$

that acts on the wave function ψ.

We thus have that in the context of quantum mechanics, dynamical symmetries can be understood as operators that act on the Hilbert space of wave functions and map the set of solutions of the time-independent Schrödinger equation onto itself. If the notion of a dynamical symmetry is understood in this manner, it is easy to show that the necessary and sufficient condition for an operator ${\hat{P}}_R$ to implement a symmetry transformation is given by the condition

(3)$\begin{array}{}\left[\hat{H},{\hat{P}}_R\right]=0.\end{array}$

In quantum mechanics, that is to say, an operator ${\hat{P}}_R$ implements a symmetry transformation if and only if it commutes with the Hamiltonian.

3.2. Conservation Laws

Equipped with this definition of symmetry, Wigner could go on to explore the consequences that symmetries have in quantum mechanics. To see that conservation laws are among these consequences we need to introduce the time-dependent Schrödinger equation,

(4)$\begin{array}{}i\hslash \frac{\partial }{\partial t}\psi \left(\overrightarrow{r},t\right)=\hat{H}\psi \left(\overrightarrow{r},t\right),\end{array}$

which describes the time evolution of an arbitrary wave function $\psi \left(\overrightarrow{r},t\right)$.

Second, and more important, it follows from (4) that symmetries and conservation laws are related to each other in quantum mechanics. If a linear operator ${\hat{P}}_R$ satisfies the commutation condition (3), its eigenstates ${\psi }_{R_1},\dots ,{\psi }_{R_n}$, with eigenvalues $a_{R_1},\dots ,a_{R_n}$, will be stationary states. The following set of conservation laws will then obtain:

(5)$\begin{array}{}\begin{array}{ccc}\frac{d}{\mathrm{dt}}|c_{R_i}\left(t\right){|}^2=0,& \mathrm{for}& i=1,\dots ,n,\end{array}\end{array}$

for the coefficients $c_{R_i}$ that result from expressing an arbitrary wave function ψ with respect to the basis provided by the eigenstates of ${\hat{P}}_R$ as

(6)$\begin{array}{}\psi =\sum _{i=0}^n{c}_{R_i}\left(t\right){\psi }_{R_i}.\end{array}$

3.3. Further Consequences of Symmetry

The above shows that there is a relation between symmetries and conservation laws in quantum mechanics. As it turns out, however, there is a second kind of consequence that symmetries have in this theory, which obtains whenever (3) is satisfied and involves the transformation properties of the energy eigenfunctions.

To see how the presence of a symmetry affects the manner in which the energy eigenstates transform, assume that a group G of operators ${\hat{P}}_R$ satisfies (3). In this case, as we have seen, the operators of the group will implement dynamical symmetry transformations that map energy eigenstates onto energy eigenstates. For a nondegenerate eigenvalue E with eigenstate ψ_E and an operator ${\hat{P}}_R\in G$, therefore, we have that

(7)$\begin{array}{}{\hat{P}}_R{\psi }_E={\alpha }_E{\psi }_E,\end{array}$

where α_E is a complex number that will in general be different for different energy eigenvalues.

The above is no longer true if the eigenvalue E is degenerate among l different eigenfunctions $\left\{{\psi }_{E_1},\dots ,{\psi }_{E_l}\right\}$. In this case, however, the linearity of the Schrödinger equation gives us the following result. Since the time-independent Schrödinger equation is linear, the eigenfunctions ${\psi }_{E_i}$ form a vector space. It must therefore be the case that

(8)$\begin{array}{}{\hat{P}}_R{\psi }_{E_i}=\sum _{j=1}^l{D}_R{\left(G\right)}_{\mathrm{ij}}{\psi }_{E_j},\end{array}$

where the coefficients D_R(G)_ij will depend on the properties of the specific operator ${\hat{P}}_R$ under consideration and on the properties of the symmetry group G more generally. As Wigner shows, indeed, the coefficients D_R(G)_ij form a representation of G.

We thus have that, whenever (3) obtains, the energy eigenfunctions transform as representations of the relevant symmetry group. This shows that, as I advanced before, conservation laws are not the only kind of consequence that symmetries have in quantum mechanics: there is a second kind of consequence that concerns the transformation properties of the energy eigenfunctions. This second kind of consequence, furthermore, was systematically deployed by Wigner and used for various practical purposes. In his work of the late 1920s and early 1930s, for example, Wigner used result (8) to substitute explicit calculations by qualitative considerations, thereby sidestepping computational difficulties in atomic physics. This fact will turn out to be important, as we will see, in interpreting Wigner’s comments in section 5.

4.1. Noether’s Theorems

After having described the manner in which symmetries and conservation laws relate to each other in quantum mechanics, we can now turn to the case of classical mechanics. My discussion will focus on the Lagrangian formalism, and it will follow the available literature on Noether’s theorems.Footnote ¹ I will follow, in particular, Katherine Brading’s and Harvey Brown’s derivation of both of Noether’s theorems and Harvey Brown’s and Peter Holland’s work on the first theorem (Brading and Brown Reference Brading, Brown, Brading and Castellani2003; Brown and Holland Reference Brown and Holland2004). My treatment of the second theorem, additionally, is based on Katherine Brading’s work on this subject (Brading Reference Brading2002, Reference Brading, Kox and Eisenstaedt2005).

Although Noether’s theorems provide us with a very general account of the consequences that symmetries have in classical mechanics, seeing how this is the case is not immediately obvious. Noether’s primary goal in proving her two theorems, in fact, was to settle a dispute about the status of the conservation of energy in general relativity (Noether Reference Noether1918; Brading Reference Brading, Kox and Eisenstaedt2005).

To see how Noether’s work relates to the topic of symmetry, consider the manner in which her two theorems are proved. Noether took the action functional of a classical field theory

(9)$\begin{array}{}S\left[{\psi }_i,{\partial }_{\mu }{\psi }_i,x^{\mu }\right]={\int }_{R}L\left({\psi }_i,{\partial }_{\mu }{\psi }_i,x^{\mu }\right){\mathrm{dx}}^4,\end{array}$

with Lagrangian density $L$, and she required that its numerical value remains invariant under joint transformations of the x ^μ and the ψ_i:

(10)$\begin{array}{}\begin{array}{c}{x}^{\mu }\to x^{\mu }+\delta x^{\mu }\\ {\psi }_i\to {\psi }_i+\delta {\psi }_i,\end{array}\end{array}$

where the δψ_i and the δx ^μ can be parametrized in terms of r continuous parameters ω^k as

(11)$\begin{array}{}\begin{array}{c}\delta x^{\mu }={\eta }_k^{\mu }\Delta {\omega }^k\\ {\delta }_0{\psi }_i={\xi }_{\mathrm{ik}}\Delta {\omega }^k.\end{array}\end{array}$

Note that the quantities Δω^k above represent infinitesimal increments of the group parameters and that δ₀ is the so-called Lie drag, which accounts for the part of the variation in the ψ_i that is not due to the variation of the independent variables x ^μ.

Noether first showed that the numerical value of the action remains invariant under transformations of the form of (10), if and only if the following r identities are satisfied, one for each parameter ω^k in (11):

(12)$\begin{array}{}\begin{array}{ccc}{E}^i{\delta }_0{\psi }_i=j_k^{\mu },& \mathrm{for}& k=1,\dots ,r,\end{array}\end{array}$

where the Eⁱ above are the so-called Euler expressions, which vanish whenever the equations of motion of their associated fields are satisfied.Footnote ² The quantities $j_k^{\mu }$ are known as “Noether currents,” and they are given by the expression

(13)$\begin{array}{}{j}_k^{\mu }=\frac{\partial L}{\partial \left({\partial }_{\mu }{\psi }_i\right)}{\xi }_{\mathrm{ik}}-L{\eta }_k^{\mu }.\end{array}$

Noether then derived two further sets of identities from (12), which correspond to her two theorems. The case of a global symmetry group with constant parameters ω^k gives us the first theorem, which consists of the following r identities:

(14)$\begin{array}{}\begin{array}{ccc}{E}^i{\xi }_{\mathrm{ik}}={\partial }_{\mu }{j}_k^{\mu },& \mathrm{for}& k=1,\dots ,r.\end{array}\end{array}$

For the case of a local symmetry in which the ω^k depend on the x ^μ one obtains the second theorem, which consists of another set of r identities, and a further result that is sometimes known as the “boundary theorem” (Brading Reference Brading, Kox and Eisenstaedt2005). Although these two results are important in their own right, I focus on the first theorem in what follows, as it will turn out to be the most important for our purposes.

The relevance of the first theorem for the topic of symmetry derives from the fact that the requirement of numerical invariance of the action functions as a sufficient condition for a group of transformations to leave the equations of motion of the relevant theory invariant. A group of transformations that leaves the numerical value of the action functional of a theory invariant, that is to say, is a symmetry group of the theory in question, in the sense of a dynamical symmetry that was introduced before. Expression (14), therefore, provides us with an account of the consequences that symmetries have in classical mechanics. If a symmetry group leaves the numerical value of the action invariant, that is to say, then the identities in (14) obtain.

4.2. Conservation Laws

To see how the above leads to the presence of a conservation law, assume that the Euler-Lagrange equations for all the fields involved in a symmetry transformation are satisfied. In this case, the left-hand side of (14) vanishes and the first theorem gives us r continuity equations for the Noether currents:

(15)$\begin{array}{}\begin{array}{ccc}{\partial }_{\mu }{j}_k^{\mu }=0,& \mathrm{for}& k=1,\dots ,r.\end{array}\end{array}$

The case of a local symmetry introduces a few nuances, but one obtains a set of continuity equations of the form of (15) also in this case (Brading Reference Brading2002).

Although the continuity equations in (15) are sometimes referred to as “local conservation laws,” I reserve the term “conservation law” for expressions that explicitly assert the invariance in time of some quantity. In order to obtain these kinds conservation laws, one may simply integrate the continuity equations in (15) over a three-dimensional volume V to obtain

(16)$\begin{array}{}\begin{array}{ccc}{\int }_V\left(\overrightarrow{\nabla }\cdot \overrightarrow{j_k}+\frac{d}{\mathrm{dt}}{j}_k^0\right){\mathrm{dx}}^3=0,& \mathrm{for}& k=1,\dots ,r.\end{array}\end{array}$

The first term in the left-hand side of (16) can be turned into a surface integral by making use of Gauss’s theorem. If the $\left|\overrightarrow{j_k}\right|$ satisfy appropriate boundary conditions, then the surface term can be made arbitrarily small for a sufficiently large volume, and we obtain r conservation laws

(17)$\begin{array}{}\begin{array}{ccc}\frac{{\mathrm{dQ}}_k}{\mathrm{dt}}=0,& \mathrm{for}& k=1,\dots ,r,\end{array}\end{array}$

for the “Noether charges” given by

(18)$\begin{array}{}{Q}_k={\int }_V{j}_k^0{\mathrm{dx}}^3.\end{array}$

4.3. Restrictions

The above shows that, just like in the case of quantum mechanics, there is a relation between symmetries and conservation laws in classical mechanics. This relation, however, is mediated by Noether’s theorems, and it will only obtain as long as the two theorems are valid. As it turns out, there are limitations to the domain of applicability of Noether’s theorems, and these impose restrictions on the circumstances under which symmetries and conservation laws are related to each other in classical mechanics. The derivation of Noether’s theorems that I provided in section 4.1 will prove useful in making sense of these restrictions.

First, keep in mind that Noether’s proof was cast in the framework of the Lagrangian formalism. Her starting point, as we saw above, was the action functional of a Lagrangian field theory. Because of this, Noether’s theorems apply to Lagrangian theories only. And since not all classical theories can be written in Lagrangian form, Noether’s use of the Lagrangian formalism constrains the domain of applicability of the two theorems. As an example, consider the case of theories with velocity-dependent forces, which are in general not amenable to a Lagrangian treatment. In these kinds of theories, as Wigner (Reference Wigner1954) pointed out, one cannot derive a conservation law from a symmetry.

Second, recall that Noether’s derivation of the two theorems proceeded by requiring that the action functional remains invariant under infinitesimal transformations of both the dependent and the independent variables of a classical field theory. Noether’s focus on infinitesimal transformations is not coincidental, however, as this is necessary for her proof to go through. Her derivation, indeed, proceeded by deploying the calculus of variations, which can only be used for transformations that can be written in infinitesimal form. Since only continuous symmetries can be expressed in this manner, discrete symmetries fall outside of the domain of applicability of Noether’s theorems. Both theorems, that is to say, remain silent about the consequences that follow from invariance of the action under discrete symmetries such as reflections.

Finally, note that the relevance of Noether’s work for the topic of symmetry derived from the fact that the requirement of invariance of the action functions as a sufficient condition for a group of transformations to count as dynamical symmetries. This condition, however, turns out to be sufficient but not necessary, and we thus have that there are symmetries that leave the equations of motion of a theory invariant but fail to leave the numerical value of its action functional unchanged. The first theorem does not apply to these kinds of symmetries, which have no conservation law attached to them. Brown and Holland (Reference Brown and Holland2004) have given examples of these kinds of symmetries, which include “rescaling” transformations that leave the equations of motion unchanged but introduce a multiplicative constant in the action.

The point is sometimes made by appealing to a difference between variational and dynamical symmetries (Olver Reference Olver1986; Brown and Holland Reference Brown and Holland2004). Variational symmetries are defined as transformations that leave the numerical value of the action invariant, while the definition of a dynamical symmetry remains as in section 3. While all variational symmetries are dynamical symmetries, the converse is not true. Using this terminology, therefore, we may say that Noether’s theorems tell us about the consequences of some but not all dynamical symmetries. They only tell us, that is to say, about the consequences of those dynamical symmetries that are variational symmetries too.

We thus see that, although there is a relation between symmetries and conservation laws in classical mechanics, this relation does not obtain for all classical theories and all symmetry transformations. It only obtains, as we have just seen, for Lagrangian theories and for continuous, variational symmetries. This supports my previous claim that, in classical mechanics, a number of restrictions exist, which limit the circumstances under which symmetries and conservation laws relate to each other.