28.1 The Complex Weyl and Cliford algebras

28.1 The Complex Weyl and Cliford algebras

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In mathematics, a “ring” is a set with addition and multiplication laws that are associative and distributive (but not necessarily commutative), and an “algebra” is a ring that is also a vector space over some field of scalars. The canonical commutation and anticommutation relations define interesting algebras, called the Weyl and Cliford algebras respectively. The case of complex numbers as scalars is simplest, so we’ll start with that, before moving on to the real number case.

28.1.1 One degree of freedom, bosonic case

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Starting with the one degree of freedom case (corresponding to two operators $Q$, $P$, which is why the notation will have a 2) we can define:

Definition (Complex Weyl algebra, one degree of freedom)

The complex Weyl algebra in the one degree of freedom case is the algebra Weyl(2, $C$) generated by the elements $1,a_B,a_B^{†}.$ , satisfying the canonical commutation relations:

$$[a_B,a_B^{†}]=1,[a_B,a_B]=[a_B^{†},a_B^{†}]=0$$

In other words, Weyl(2, $C$) is the algebra one gets by taking arbitrary products and complex linear combinations of the generators. By repeated use of the commutation relation

$$a_B{a}_B^{†}=1+a_B^{†}{a}_B$$

any element of this algebra can be written as a sum of elements in normal order, of the form

$$c_{l,m}(a_B^{†}{)}^l{a}_B^m$$

with all annihilation operators $a_B$ on the right, for some complex constants $c_{l,m}.$ As a vector space over $\mathbf{C},\mathrm{Weyl}(2,\mathbf{C})$ is infinite dimensional, with a basis

$$1,a_B,a_B^{†},a_B^2,a_B^{†}{a}_B,(a_B^{†}{)}^2,a_B^3,a_B^{†}{a}_B^2,(a_B^{†}{)}^2{a}_B,(a_B^{†}{)}^3,\dots$$

This algebra is isomorphic to a more familiar one. Setting

$$a_B^{†}=z,a_B=\frac{d}{dz}$$

one sees that $\mathrm{Weyl}(2,\mathbf{C})$ can be identified with the algebra of polynomial coefficient diferential operators on functions of a complex variable $z$. As a complex vector space, the algebra is infinite dimensional, with a basis of elements

$$z^l\frac{d^m}{d{z}^m}$$

In our study of quantization by the Bargmann-Fock method, we saw that the subset of such operators consisting of complex linear combinations of

$$1,z,\frac{d}{dz},z^2,\frac{d^2}{d{z}^2},z\frac{d}{dz}$$

is closed under commutators, and is a representation of a Lie algebra of complex dimension 6. This Lie algebra includes as subalgebras the Heisenberg Lie algebra ${\mathfrak{h}}_3\otimes \mathbf{C}$ (first three elements) and the Lie algebra $\mathfrak{sl}(2,\mathbf{C})=\mathfrak{sl}(2,\mathbf{R})\otimes \mathbf{C}$ (last three elements). Note that here we are allowing complex linear combinations, so we are getting the complexification of the real six dimensional Lie algebra that appeared in our study of quantization.

Since the $a_B$ and $a_B^{†}$ are defined in terms of $P$ and $Q.$ one could of course also define the Weyl algebra as the one generated by $1,P,Q$ with the Heisenberg commutation relations, taking complex linear combinations of all products of these operators.

28.1.2 One degree of freedom, fermionic case

Changing commutators to anticommutators, one gets a diferent algebra, the Cliford algebra:

Definition (Complex Cliford algebra, one degree of freedom)

The complex Cliford algebra in the one degree of freedom case is the algebra $\mathrm{Cliff}(2,\mathbf{C})$ generated by the elements $1,a_F,a_F^{†}$ , subject to the canonical anticommutation relations $(CAR)$

$$[a_F,a_F^{†}{]}_{+}=1,[a_F,a_F{]}_{+}=[a_F^{†},a_F^{†}{]}_{+}=0$$

This algebra is a four dimensional algebra over $C$, with basis

$$1,a_F,a_F^{†},a_F^{†}{a}_F$$

since higher powers of the operators vanish, and the anticommutation relation between $a_F$ and $a_F^{†}$ can be used to normal order and put factors of $a_F$ on the right. We saw in chapter 27 that this algebra is isomorphic with the algebra $M(2,\mathbf{C})$ of 2 by $2$ complex matrices, using

$$1\leftrightarrow \left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right),a_F\leftrightarrow \left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right),a_F^{†}\leftrightarrow \left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right),a_F^{†}{a}_F\leftrightarrow \left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right)\text{\hspace{1em}(28.1)}$$

We will see in chapter 30 that there is also a way of identifying this algebra with “diferential operators in fermionic variables”, analogous to what happens in the bosonic (Weyl algebra) case.

Recall that the bosonic annihilation and creation operators were originally defined in terms of the $P$ and $Q$ operators by

$$a_B=\frac{1}{\sqrt{2}}(Q+iP),a_B^{†}=\frac{1}{\sqrt{2}}(Q-iP)$$

Looking for the fermionic analogs of the operators $Q$ and $P,$ we use a slightly diferent normalization, and set

$$a_F=\frac{1}{2}({\gamma }_1+i{\gamma }_2),a_F^{†}=\frac{1}{2}({\gamma }_1-i{\gamma }_2)$$

so

$${\gamma }_1=a_F+a_F^{†},{\gamma }_2=\frac{1}{i}(a_F-a_F^{†})$$

and the CAR imply that the operators ${\gamma }_j$ satisfy the anticommutation relations

$$\begin{array}{rl}[{\gamma }_1,{\gamma }_1{]}_{+}& =[a_F+a_F^{†},a_F+a_F^{†}{]}_{+}=2\\ [{\gamma }_2,{\gamma }_2{]}_{+}& =-[a_F-a_F^{†},a_F-a_F^{†}{]}_{+}=2\\ [{\gamma }_1,{\gamma }_2{]}_{+}& =\frac{1}{i}[a_F+a_F^{†},a_F-a_F^{†}{]}_{+}=0\end{array}$$

From this we see that

• One could alternatively have defined Clif(2, $C$) as the algebra generated by $1,{\gamma }_1,{\gamma }_2$ , subject to the relations

$$[{\gamma }_j,{\gamma }_k{]}_{+}=2{\delta }_{jk}$$

• Using just the generators 1 and ${\gamma }_1$ , one gets an algebra $\mathrm{Cliff}(1,\mathbf{C})$ , generated by $1,{\gamma }_1$ , with the relation

$${\gamma }_1^2=1$$

This is a two dimensional complex algebra, isomorphic to $\mathbf{C}\oplus \mathbf{C}$

28.1.3 Multiple degrees of freedom

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For a larger number of degrees of freedom, one can generalize the above and define Weyl and Cliford algebras as follows:

Definition (Complex Weyl algebras)

The complex Weyl algebra for $d$ degrees of freedom is the algebra $\mathrm{Weyl}(2d,\mathbf{C})$ generated by the elements 1, $,a_{Bj},a_{B_j^{†}}$ 2 $j=1,\dots ,d$ satisfying the CCR

$$[a_{Bj},a_{Bk}^{†}]={\delta }_{jk}1,[a_{Bj},a_{Bk}]=[a_{Bj}^{†},a_{Bk}^{†}]=0$$

Weyl(2d, $C$) can be identified with the algebra of polynomial coeficient differential operators in $d$ complex variables $z_1,z_2,\dots ,z_d$ . The subspace of complex linear combinations of the elements

$$1,z_j,\frac{\partial }{\partial z_j},z_j{z}_k,\frac{{\partial }^2}{\partial z_j\partial z_k},z_j\frac{\partial }{\partial z_k}$$

is closed under commutators and provides a representation of the complexification of the Lie algebra ${\mathfrak{h}}_{2d+1}⋊\mathfrak{sp}(2d,\mathbf{R})$ built out of the Heisenberg Lie algebra for $d$ degrees of freedom and the Lie algebra of the symplectic group $Sp(2d,\mathbf{R})$ . Recall that this is the Lie algebra of polynomials of degree at most 2 on the phase space ${\mathbf{R}}^{2d}$ , with the Poisson bracket as Lie bracket. The complex Weyl algebra could also be defined by taking complex linear combinations of products of generators $1,P_j,Q_j$ , subject to the Heisenberg commutation relations.

For Cliford algebras one has:

Definition (Complex Cliford algebras, using annihilation and creation operators)

The complex Cliford algebra for $d$ degrees of freedom is the algebra $\mathrm{Cliff}(2d,\mathbf{C})$ generated by $1,a_{Fj},a_{Fj}†forj=1,2,...,d$ satisfying the CAR

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

or, alternatively, one has the following more general definition that also works in the odd dimensional case:

Definition (Complex Cliford algebras)

The complex Cliford algebra in $n$ variables is the algebra $\mathrm{Cliff}(n,\mathbf{C})$ generated by 1, ${\gamma }_j$ for $j=1,2,\dots ,n$ satisfying the relations

$$[{\gamma }_j,{\gamma }_k{]}_{+}=2{\delta }_{jk}$$

We won’t try and prove this here, but one can show that, abstractly as algebras, the complex Cliford algebras are something well known. Generalizing the case $d=1$ where we saw that Clif(2, $C$) was isomorphic to the algebra of 2 by 2 complex matrices, one has isomorphisms

$$\mathrm{Cliff}(2d,\mathbf{C})\leftrightarrow M(2^d,\mathbf{C})$$

in the even dimensional case, and in the odd dimensional case

$$\mathrm{Cliff}(2d+1,\mathbf{C})\leftrightarrow M(2^d,\mathbf{C})\oplus M(2^d,\mathbf{C})$$

Two properties of $\mathrm{Cliff}(n,\mathbf{C})$ are

• As a vector space over $C$, a basis of $\mathrm{Cliff}(n,\mathbf{C})$ is the set of elements

$$1,{\gamma }_j,{\gamma }_j{\gamma }_k,{\gamma }_j{\gamma }_k{\gamma }_l,\dots ,{\gamma }_1{\gamma }_2{\gamma }_3\dots {\gamma }_{n-1}{\gamma }_n$$

for indices $j,k,l,\cdot \cdot \cdot \in 1,2,\dots ,n$ , with $j<k<l<\cdots$ . To show this, consider all products of the generators, and use the commutation relations for the ${\gamma }_j$ to identify any such product with an element of this basis. The relation ${\dot{\gamma }}_j^2=1$ shows that repeated occurrences of a ${\gamma }_j$ can be removed. The relation ${\gamma }_j{\gamma }_k=-{\gamma }_k{\gamma }_j$ can then be used to put elements of the product in the order of a basis element as above.

• As a vector space over $C$, Clif(n, $C$) has dimension $2^n$ . One way to see this is to consider the product

$$(1+{\gamma }_1)(1+{\gamma }_2)\dots (1+{\gamma }_n)$$

which will have $2^n$ terms that are exactly those of the basis listed above.

Chapter contents · Next section: 28.2 Real Cliford algebras · 中文版本

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