27.2 Multiple degrees of freedom

27.2 Multiple degrees of freedom

Concept links · terms present in this machine draft; source roles are unverified: eigenvalue · Lie algebra · Lie algebra representation · Pauli matrices

For the case of $d$ degrees of freedom, one has this variant of the canonical commutation relations (CCR) amongst the bosonic annihilation and creation operators $a_{Bj}$ and $a_{B_j^{†}}$ :

Definition (Canonical anticommutation relations)

A set of 2d operators

$$a_{Fj},a_{Fj}^{†},j=1,\dots ,d$$

is said to satisfy the canonical anticommutation relations $(CAR)$ when one has

$$[a_{Fj},a_{Fk}^{†}{]}_{+}={\delta }_{jk}1,[a_{Fj},a_{Fk}{]}_{+}=0,[a_{Fj}^{†},a_{Fk}^{†}{]}_{+}=0$$

In this case one may choose as the state space the tensor product of N copies of the single fermionic oscillator state space

$${\mathcal{H}}_F=({\mathbf{C}}^2{)}^{\otimes d}=\underset{d\mathrm{times}}{\underbrace{{\mathbf{C}}^2\otimes {\mathbf{C}}^2\otimes \cdots \otimes {\mathbf{C}}^2}}$$

The dimension of ${\mathcal{H}}_F$ will be $2^d$ . On this space an explicit construction of the operators ${\mathbf{a}}_{F_j}$ and $a_{F_j^{†}}$ in terms of Pauli matrices is

$$a_{Fj}=\underset{j-1\text{ times }}{\underbrace{{\sigma }_3\otimes {\sigma }_3\otimes \cdots \otimes {\sigma }_3}}\otimes \left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right)\otimes 1\otimes \cdots \otimes 1$$ $$a_{Fj}^{†}=\underset{j-1\text{ times }}{\underbrace{{\sigma }_3\otimes {\sigma }_3\otimes \cdots \otimes {\sigma }_3}}\otimes \left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)\otimes 1\otimes \cdots \otimes 1$$

The factors of ${\sigma }_3$ are there as one possible way to ensure that

$$[a_{Fj},a_{Fk}{]}_{+}=[a_{Fj}^{†},a_{Fk}^{†}{]}_{+}=[a_{Fj},a_{Fk}^{†}{]}_{+}=0$$

are satisfied for $j\ne k$ since then one will get in the tensor product factors

$$[{\sigma }_3,\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right){]}_{+}=0\text{or}[{\sigma }_3,\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right){]}_{+}=0$$

While this sort of tensor product construction is useful for discussing the physics of multiple qubits, in general it is easier to not work with large tensor products, and the Cliford algebra formalism we will describe in chapter 28 avoids this.

The number operators will be

$$N_{Fj}=a_{Fj}^{†}{a}_{Fj}$$

These will commute with each other, so can be simultaneously diagonalized, with eigenvalues $n_j=0,1$ 1. One can take as a basis of ${\mathcal{H}}_F$ the $2^d$ states

$$|n_1,n_2,\dots ,n_d⟩$$

which are the natural basis states for $({\mathbf{C}}^2{)}^{\otimes d}$ given by $d$ choices of either |0⟩ or |1⟩.

As an example, for the case $d=3$ the picture shows the pattern of states and their energy levels for the bosonic and fermionic cases. In the bosonic case the lowest energy state is at positive energy and there are an infinite number of states of ever increasing energy. In the fermionic case the lowest energy state is at negative energy, with the pattern of energy eigenvalues of the finite number of states symmetric about the zero energy level.

Figure 27.1: $N=3$ oscillator energy eigenstates.

Just as in the bosonic case, we can consider quadratic combinations of creation and annihilation operators of the form

$$U_A^{\prime }=\sum _{j,k}{a}_{Fj}^{†}{A}_{jk}{a}_{Fk}$$

and we have

Theorem 27.1

For $A\in \mathfrak{gl}(d,\mathbf{C})$ a d by $d$ complex matrix one has

$$[U_A^{\prime },U_{A^{\prime }}^{\prime }]=U_{[A,A^{\prime }]}$$

So

$$A\in \mathfrak{gl}(d,\mathbf{C})\to U_A^{\prime }$$

is a Lie algebra representation of $\mathfrak{gl}(d,\mathbf{C})$ on ${\mathcal{H}}_F$

One also has (for column vectors ${\mathbf{a}}_F$ with components $a_{F1},\dots ,a_{Fd})$

$$[U_A^{\prime },{\mathbf{a}}_F^{†}]=A^T{\mathbf{a}}_F^{†},[U_A^{\prime },{\mathbf{a}}_F]=-A{\mathbf{a}}_F\text{\hspace{1em}(27.1)}$$

Proof. The proof is similar to that of 25.1, except besides the relation

$$[AB,C]=A[B,C]+[A,C]B$$

we also use the relation

$$[AB,C]=A[B,C{]}_{+}-[A,C{]}_{+}B$$

For example

$$\begin{array}{rl}[U_A^{\prime },a_{Fl}^{†}]& =\sum _{j,k}[a_{Fj}^{†}{A}_{jk}{a}_{Fk},a_{Fl}^{†}]\\ & =\sum _{j,k}{a}_{Fj}^{†}{A}_{jk}[a_{Fk},a_{Fl}^{†}{]}_{+}\\ & =\sum _j{a}_{Fj}^{†}{A}_{jl}\end{array}$$

The Hamiltonian is

$$H=\sum _j(N_{Fj}-\frac{1}{2}1)$$

which (up to the constant $\begin{array}{l}\frac{1}{2}\end{array}$ that doesn’t contribute to commutation relations) is just $U_B^{\prime }$ for the case $B=1$ . Since this commutes with all other d by $d$ matrices, we have

$$[H,U_A^{\prime }]=0$$

for all $A\in \mathfrak{gl}(d,\mathbf{C})$ , so these are symmetries and we have a representation of the Lie algebra $\mathfrak{gl}(d,\mathbf{C})$ on each energy eigenspace. Only for $A\in \mathfrak{u}(d)$ (A a skew-adjoint matrix) will the representation turn out to be unitary.

Previous section: 27.1 Canonical anticommutation relations and the fermionic oscillator · Next section: 27.3 For further reading · 中文版本

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正文：英文 · OCR 机器稿 · 待校对

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原书 PDF · 印刷页 305、306、307、308、309、310

来源版本：2025-10-20

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