38.2 Internal symmetries

38.2 Internal symmetries

Concept links · terms present in this machine draft; source roles are unverified: group action

Since the phase space ${\mathcal{H}}_1$ is a space of complex functions, there is an obvious group that acts unitarily on this space: the group $U(1)$ of phase transformations of the complex values of the function. Such a group action that acts trivially on the spatial coordinates but non-trivially on the values of $\psi (x)$ is called an “internal symmetry”. If the fields $\psi$ have multiple components, taking values in ${\mathbf{C}}^n$ , there will be a unitary action of the larger group $U(n)$ .

38.2.1 U(1) symmetry

Concept links · terms present in this machine draft; source roles are unverified: eigenvalue · irreducible representation · U(1)

In chapter 2 we saw that the fact that irreducible representations of $U(1)$ are labeled by integers is responsible for the term “quantization”: since quantum states are representations of this group, they break up into states characterized by integers, with these integers counting the number of “quanta”. In the nonrelativistic quantum field theory, this integer will be the total particle number. Such a theory can be thought of as a harmonic oscillator with an infinite number of degrees of freedom, and the total particle number is the total occupation number, summed over all degrees of freedom.

Consider the $U(1)$ action on the fields $\Psi (x),\overline{\Psi }(x)$ given by

$$\Psi (x)\to e^{-i\theta }\Psi (x),\overline{\Psi }(x)\to e^{i\theta }\overline{\Psi }(x)\text{\hspace{1em}(38.1)}$$

This is an infinite dimensional generalization of the case worked out in section 24.2, where recall that the moment map was $\mu =\overline{z}z$ and

$$\{z\overline{z},\overline{z}\}=i\overline{z}\{z\overline{z},z\}=-iz$$

There were two possible choices for the unitary operator that will be the quantization of $z\overline{z}:$

$$z\overline{z}\to -\frac{i}{2}(a^{†}a+a{a}^{†})$$

This will have eigenvalues $-i(n+\frac{1}{2}),n=0,1,2\dots$

$$z\overline{z}\to -i{a}^{†}a$$

This is the normal ordered form, with eigenvalues −in.

With either choice, we get a number operator

$$N=\frac{1}{2}(a^{†}a+a{a}^{†}),\mathrm{or}N=\frac{1}{2}:(a^{†}a+a{a}^{†}):=a^{†}a$$

In both cases we have

$$[N,a]=-a,[N,a^{†}]=a^{†}$$

so

$$e^{i\theta N}a{e}^{-i\theta N}=e^{-i\theta }a,e^{i\theta N}{a}^{†}{e}^{-i\theta N}=e^{i\theta }{a}^{†}$$

Either choice of N will give the same action on operators. However, on states only the normal ordered one will have the desirable feature that

$$N|0⟩=0,e^{iN\theta }|0⟩=|0⟩$$

Since we now want to treat fields, adding together an infinite number of such oscillator degrees of freedom, we will need the normal ordered version in order to not get $\propto \cdot :\frac{1}{2}$ as the number eigenvalue for the vacuum state.

We now generalize as described in section 38.1 and $\mathrm{get}$ , in momentum space, the expression

$$\hat{N}={\int }_{-\infty }^{+\infty }{a}^{†}(p)a(p)dp\text{\hspace{1em}(38.2)}$$

which is just the number operator already discussed in chapter 36. Recall from section 36.3 that this sort of operator product requires some interpretation in order to give it a well-defined meaning, either as a limit of a finite dimensional definition, or by giving it a distributional interpretation.

Fourier transforming to position space, one can work with $\hat{\Psi }(x),{\hat{\Psi }}^{†}(x)$ instead of $a(p),a^{†}(p)$ and find that

$$\hat{N}={\int }_{-\infty }^{+\infty }{\hat{\Psi }}^{†}(x)\hat{\Psi }(x)dx\text{\hspace{1em}(38.3)}$$

${\hat{\Psi }}^{†}(x)\hat{\Psi }(x)$ can be interpreted as an operator-valued distribution, with the physical interpretation of measuring the number density at x. On field operators, $\hat{N}$ satisfies

$$[\hat{N},\hat{\Psi }]=-\hat{\Psi },[\hat{N},{\hat{\Psi }}^{†}]={\hat{\Psi }}^{†}$$

so $\hat{\Psi }$ acts on states by reducing the eigenvalue of $\hat{N}$ by one, while ${\hat{\Psi }}^{†}$ acts on states by increasing the eigenvalue of $\stackrel{\approx }{N}$ by one. Exponentiating gives

$$e^{i\theta \hat{N}}\hat{\Psi }{e}^{-i\theta \hat{N}}=e^{-i\theta }\hat{\Psi },e^{i\theta \hat{N}}{\hat{\Psi }}^{†}{e}^{-i\theta \hat{N}}=e^{i\theta }{\hat{\Psi }}^{†}$$

which are the quantized versions of the $U(1)$ action on the phase space coordinates (see equations 38.1) that we began our discussion with.

An important property of $\hat{N}$ that can be straightforwardly checked is that

$$[\hat{N},\hat{H}]=\left[\hat{N},{\int }_{-\infty }^{+\infty }{\hat{\Psi }}^{†}(x)\frac{-1}{2m}\frac{{\partial }^2}{\partial x^2}\hat{\Psi }(x)dx\right]=0$$

This implies that particle number is a conserved quantity: if we start out with a state with a definite particle number, this will remain constant. Note that the origin of this conservation law comes from the fact that $\hat{N}$ is the quantized generator of the $U(1)$ symmetry of phase transformations on complex-valued fields $\Psi$. If we start with any Hamiltonian function $h$ on ${\mathcal{H}}_1$ that is invariant under the $U(1)$ (i.e., built out of terms with an equal number of $\Psi$s and $\overline{\Psi }\mathrm{s})$ then for such a theory $\hat{N}$ will commute with $\hat{H}$ and particle number will be conserved.

38.2.2 U(n) symmetry

Concept links · terms present in this machine draft; source roles are unverified: adjoint operator · Lie algebra · Lie algebra representation · unitary representation · irreducible representation · symmetry group · Lie bracket

By taking fields with values in ${\mathbf{C}}^n$ , or, equivalently, $n$ diferent species of complex-valued field ${\Psi }_j,j=1,2,\dots ,n.$ , quantum field theories with larger internal symmetry groups than $U(1)$ can easily be constructed. Taking as Hamiltonian function

$$h={\int }_{-\infty }^{+\infty }\sum _{j=1}^n{\Psi }_j(x)\frac{-1}{2m}\frac{{\partial }^2}{\partial x^2}{\overline{\Psi }}_j(x)dx\text{\hspace{1em}(38.4)}$$

gives a Hamiltonian that will be invariant not just under $U(1)$ phase transformations, but also under transformations

$$\left(\begin{array}{c}{\Psi }_1\\ {\Psi }_2\\ ⋮\\ {\Psi }_n\end{array}\right)\to U\left(\begin{array}{c}{\Psi }_1\\ {\Psi }_2\\ ⋮\\ {\Psi }_n\end{array}\right)$$

where $U$ is an n by $n$ unitary matrix. The Poisson brackets will be

$$\{{\Psi }_j(x),{\overline{\Psi }}_k(x^{\prime })\}=i\delta (x-x^{\prime }){\delta }_{jk}$$

and are also invariant under such transformations by $U\in U(n)$

As in the $U(1)$ case, we begin by considering the case of one particular value of $p$ or of x, for which the phase space is ${\mathbf{C}}^n$ , with coordinates $z_j,{\bar{z}}_j$ . As we saw in section 25.2, the $n^2$ quadratic combinations $z_j{\overline{z}}_k$ for $j=1,\dots ,n,k=1,\dots ,n$ will generalize the role played by zz in the $n=1$ case, with their Poisson bracket relations exactly the Lie bracket relations of the Lie algebra $\mathfrak{u}(n)$ (or, considering all complex linear combinations, $\mathfrak{gl}(n,\mathbf{C}))$ .

After quantization, these quadratic combinations become quadratic combinations of annihilation and creation operators $a_j,a_j^{†}$ satisfying

$$[a_j,a_k^{†}]={\delta }_{jk}$$

Recall (theorem 25.2) that for n by $n$ matrices $X$ and $Y$

$$\left[\sum _{j,k=1}^n{a}_j^{†}{X}_{jk}{a}_k,\sum _{j,k=1}^n{a}_j^{†}{Y}_{jk}{a}_k\right]=\sum _{j,k=1}^n{a}_j^{†}[X,Y{]}_{jk}{a}_k$$

$\mathrm{So}$ , for each $X$ in the Lie algebra $\mathfrak{gl}(n,\mathbf{C})$ , quantization will give us a representation of $\mathfrak{gl}(n,\mathbf{C})$ where $X$ acts as the operator

$$\sum _{j,k=1}^n{a}_j^{†}{X}_{jk}{a}_k$$

When the matrices $X$ are chosen to be skew-adjoint $(X_{jk}=-\overline{X_{kj}})$ this construction will give us a unitary representation of $\mathfrak{u}(n)$

As in the $U(1)$ case, one gets an operator in the quantum field theory by integrating over quadratic combinations of the $a(p),a^{†}(p)$ in momentum space, or the field operators $\hat{\Psi }(x),{\hat{\Psi }}^{†}(x)$ in configuration space, finding for each $X\in$ $\mathfrak{u}(n)$ an operator

$$\hat{X}={\int }_{-\infty }^{+\infty }\sum _{j,k=1}^n{\hat{\Psi }}_j^{†}(x)X_{jk}{\hat{\Psi }}_k(x)dx={\int }_{-\infty }^{+\infty }\sum _{j,k=1}^n{a}_j^{†}(p)X_{jk}{a}_k(p)dp\text{\hspace{1em}(38.5)}$$

This satisfies

$$[\hat{X},\hat{Y}]=\widehat{[X,Y]}\text{\hspace{1em}(38.6)}$$

and, acting on operators

$$[\hat{X},{\hat{\Psi }}_j(x)]=-\sum _{k=1}^n{X}_{jk}{\hat{\Psi }}_k(x),[\hat{X},{\hat{\Psi }}_j^{†}(x)]=\sum _{k=1}^n{X}_{kj}{\hat{\Psi }}_k^{†}(x)\text{\hspace{1em}(38.7)}$$

$\hat{X}$ provides a Lie algebra representation of $\mathfrak{u}(n)$ on the multi-particle state space. After exponentiation, this representation takes

$$e^X\in U(n)\to U(e^X)=e^{\hat{X}}=e^{{\int }_{-\infty }^{+\infty }{\sum }_{b,c=1}^n{\hat{\Psi }}_j^{†}(x)X_{jk}{\hat{\Psi }}_k(x)dx}$$

The construction of the operator $\hat{X}$ above is an infinite dimensional example of our standard method of creating a Lie algebra representation by quantizing moment map functions. In this case the quadratic moment map function on the space of solutions of the Schr¨odinger equation is

$${\mu }_X=i{\int }_{-\infty }^{+\infty }\sum _{j,k=1}^n{\Psi }_j(x)X_{jk}{\overline{\Psi }}_k(x)dx$$

which (generalizing the finite dimensional case of theorem 25.1) satisfies the Poisson bracket relations

$$\{{\mu }_X,{\mu }_Y\}={\mu }_{[X,Y]}$$ $$\{{\mu }_X,{\overline{\Psi }}_j(x)\}=-X_{jk}{\overline{\Psi }}_j(x),\{{\mu }_X,{\Psi }_j(x)\}=X_{kj}{\Psi }_k(x)$$

After quantization these become the operator relations 38.6 and 38.7. Note that the factor of $i$ in the expression for ${\mu }_X$ is there to make it a real function for $X\in$ $\mathfrak{u}(n)$ . Quantization of this would give a self-adjoint operator, so multiplication by −$i$ makes the expression for $\hat{X}$ skew-adjoint, and thus a unitary Lie algebra representation.

When, as for the free particle case of equation $38.4,$ , the Hamiltonian is invariant under $U(n)$ transformations of the fields ${\Psi }_j,{\overline{\Psi }}_j$ , then we will have

$$[\hat{X},\hat{H}]=0$$

Energy eigenstates in the multi-particle state space will break up into irreducible representations of $U(n)$ and can be labeled accordingly.

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