31.1 Quantization of pseudo-classical systems

31.1 Quantization of pseudo-classical systems

Concept links · terms present in this machine draft; source roles are unverified: vector space · linear map · Lie algebra · Lie algebra representation · unitary representation · irreducible representation · Lie group · complexification

In the bosonic case, quantization was based on finding a representation of the Heisenberg Lie algebra of linear functions on phase space, or more explicitly, for basis elements $q_j,p_j$ of this Lie algebra finding operators $Q_j,P_j$ satisfying the Heisenberg commutation relations. In the fermionic case, the analog of the Heisenberg Lie algebra is not a Lie algebra, but a Lie superalgebra, with basis elements $1,{\xi }_j,j=1,\dots ,n$ and a Lie superbracket given by the fermionic

Poisson bracket, which on basis elements is

$$\{{\xi }_j,{\xi }_k{\}}_{+}=\pm {\delta }_{jk},\{{\xi }_j,1{\}}_{+}=0,\{1,1{\}}_{+}=0$$

Quantization is given by finding a representation of this Lie superalgebra. The definition of a Lie algebra representation can be generalized to that of a Lie superalgebra representation by:

Definition (Representation of a Lie superalgebra)

A representation of a Lie superalgebra is a homomorphism $\Phi$ preserving the superbracket

$$[\Phi (X),\Phi (Y){]}_{\pm }=\Phi ([X,Y{]}_{\pm })$$

This takes values in a Lie superalgebra of linear operators, with $|\Phi (X)|=|X|$ and

$$[\Phi (X),\Phi (Y){]}_{\pm }=\Phi (X)\Phi (Y)-(-{)}^{|X||Y|}\Phi (Y)\Phi (X)$$

A representation of the pseudo-classical Lie superalgebra (and thus a quantization of the pseudo-classical system) will be given by finding a linear map ${\Gamma }^{+}$ that takes basis elements ${\xi }_j$ to operators ${\Gamma }^{+}({\xi }_j)$ satisfying the relations

$$[{\Gamma }^{+}({\xi }_j),{\Gamma }^{+}({\xi }_k){]}_{+}=\pm {\delta }_{jk}{\Gamma }^{+}(1),[{\Gamma }^{+}({\xi }_j),{\Gamma }^{+}(1)]=[{\Gamma }^{+}(1),{\Gamma }^{+}(1)]=0$$

These relations can be satisfied by taking

$${\Gamma }^{+}({\xi }_j)=\frac{1}{\sqrt{2}}{\gamma }_j,{\Gamma }^{+}(1)=1$$

since then

$$[{\Gamma }^{+}({\xi }_j),{\Gamma }^{+}({\xi }_k){]}_{+}=\frac{1}{2}[{\gamma }_j,{\gamma }_k{]}_{+}=\pm {\delta }_{jk}$$

are exactly the Cliford algebra relations. This can be extended to a representation of the functions of the ${\xi }_j$ of order two or less by

Theorem

A representation of the Lie superalgebra of anticommuting functions of coordinates ${\xi }_j$ on ${\mathbf{R}}^n$ of order two or less is given by

$${\Gamma }^{+}(1)=1,{\Gamma }^{+}({\xi }_j)=\frac{1}{\sqrt{2}}{\gamma }_j,{\Gamma }^{+}({\xi }_j{\xi }_k)=\frac{1}{2}{\gamma }_j{\gamma }_k$$

Proof. We have already seen that this is a representation for polynomials in ${\xi }_j$ of degree zero and one. For simplicity just considering the case $s=0$ (positive definite inner product), in degree two the fermionic Poisson bracket relations are given by equations 30.1 and 30.2. For 30.1, one can show that the products of Cliford algebra generators

$${\Gamma }^{+}({\xi }_j{\xi }_k)=\frac{1}{2}{\gamma }_j{\gamma }_k$$

satisfy

$$\left[\frac{1}{2}{\gamma }_j{\gamma }_k,{\gamma }_l\right]={\delta }_{kl}{\gamma }_j-{\delta }_{jl}{\gamma }_k$$

by using the Cliford algebra relations, or by noting that this is the special case of equation 29.5 for $\mathbf{v}={\mathbf{e}}_l$ . That equation shows that commuting by $-\frac{1}{2}{\gamma }_j{\gamma }_k$ 2 acts by the infinitesimal rotation ${ϵ}_{jk}$ in the $jk$ coordinate plane.

For 30.2, the Cliford algebra relations can again be used to show

$$\left[\frac{1}{2}{\gamma }_j{\gamma }_k,\frac{1}{2}{\gamma }_l{\gamma }_m\right]={\delta }_{kl}\frac{1}{2}{\gamma }_j{\gamma }_m-{\delta }_{jl}\frac{1}{2}{\gamma }_k{\gamma }_m+{\delta }_{km}\frac{1}{2}{\gamma }_l{\gamma }_j-{\delta }_{jm}\frac{1}{2}{\gamma }_l{\gamma }_k$$

One could instead use the commutation relations for the $\mathfrak{so}(n)$ Lie algebra satisfied by the basis elements ${ϵ}_{jk}$ corresponding to infinitesimal rotations. One must get identical commutation relations for the $-\frac{1}{2}{\gamma }_j{\gamma }_k$ and can show that these are the relations needed for commutators of ${\Gamma }^{+}\overline{(}{\xi }_j{\xi }_k)$ and ${\Gamma }^{+}$($\xi$ $\xi$ ).

Note that here we are not introducing the factors of i into the definition of quantization that in the bosonic case were necessary to get a unitary representation of the Lie group corresponding to the real Heisenberg Lie algebra ${\mathfrak{h}}_{2d+1}$ In the bosonic case we worked with all complex linear combinations of powers of the $Q_j,P_j$ (the complex Weyl algebra $\mathrm{Weyl}(2d,\mathbf{C}))$ ), and thus had to identify the specific complex linear combinations of these that gave unitary representations of the Lie algebra $h_{2d+1}⋊\mathfrak{sp}(2d,\mathbf{R})$ . Here we are not complexifying for now, but working with the real Cliford algebra $\mathrm{Cliff}(r,s,\mathbf{R})$ , and it is the irreducible representations of this algebra that provide an analog of the unique interesting irreducible representation of ${\mathfrak{h}}_{2d+1}$ . In the Cliford algebra case, the representations of interest are not just Lie algebra representations and may be on real vector spaces. There is no analog of the unitarity property of the ${\mathfrak{h}}_{2d+1}$ representation.

In the bosonic case we found that $Sp(2d,\mathbf{R})$ acted on the bosonic dual phase space, preserving the antisymmetric bilinear form Ω that determined the Lie algebra ${\mathfrak{h}}_{2d+1}$ , so it acted on this Lie algebra by automorphisms. We saw (see chapter 20) that intertwining operators there gave us a representation of the double cover of $Sp(2d,\mathbf{R})$ (the metaplectic representation), with the Lie algebra representation given by the quantization of quadratic functions of the $q_j,p_j$ phase space coordinates. There is a closely analogous story in the fermionic case, where $SO(r,s,\mathbf{R})$ acts on the fermionic phase space $V,$ , preserving the symmetric bilinear form $(\cdot ,\cdot )$ that determines the Cliford algebra relations. Here a representation of the spin group $Spin(r,s,\mathbf{R})$ double covering $SO(r,s,\mathbf{R})$ is constructed using intertwining operators, with the Lie algebra representation given by quadratic combinations of the quantizations of the fermionic coordinates ${\xi }_j$ . The case of $r=3,s=1$ will be of importance later in our discussion of special relativity (see chapter 41), giving the spinor representation of the Lorentz group.

The fermionic analog of 20.1 is

$$U_k{\Gamma }^{+}(\xi )U_k^{-1}={\Gamma }^{+}({\varphi }_{k_0}(\xi ))\text{\hspace{1em}(31.1)}$$

Here $k_0\in SO(r,s,\mathbf{R}),\xi \in V^{\ast }={\mathbf{R}}^n(n=r+s),{\varphi }_{k_0}$ is the action of $k_0$ on $V^{\ast }$ . The $U_k$ for $k={\Phi }^{-1}(k_0)\in Spin(r,s)(\Phi$ is the 2-fold covering map) are the intertwining operators we are looking for. The fermionic analog of 20.2 is

$$[U_L^{\prime },{\Gamma }^{+}(\xi )]={\Gamma }^{+}(L\cdot \xi )$$

where $L\in \mathfrak{so}(r,s,\mathbf{R})$ and $L$ acts on $V^{\ast }$ as an infinitesimal orthogonal transformation. In terms of basis vectors of $V^{\ast }$

$$\mathbf{\xi }=\left(\begin{array}{c}{\xi }_1\\ ⋮\\ {\xi }_n\end{array}\right)$$

this says

$$[U_L^{\prime },{\Gamma }^{+}(\xi )]={\Gamma }^{+}(L^T\xi )$$

Just as in the bosonic case, the $U_L^{\prime }$ can be found by looking first at the pseudo-classical case, where one has theorem 30.1 which says

$$\{{\mu }_L,\xi {\}}_{+}=L^T\xi$$

where

$${\mu }_L=\frac{1}{2}\mathbf{\xi }\cdot L\mathbf{\xi }=\frac{1}{2}\sum _{j,k}{L}_{jk}{\xi }_j{\xi }_k$$

One then takes

$$U_L^{\prime }={\Gamma }^{+}({\mu }_L)=\frac{1}{4}\sum _{j,k}{L}_{jk}{\gamma }_j{\gamma }_k$$

For the positive definite case $s=0$ and a rotation in the $jk$ plane, with $L={ϵ}_{jk}$ one recovers formulas 29.4 and 29.5 from chapter 29, with

$$\left[-\frac{1}{2}{\gamma }_j{\gamma }_k,\gamma (\mathbf{v})\right]=\gamma ({ϵ}_{jk}\mathbf{v})$$

the infinitesimal action of a rotation on the $\gamma$ matrices, and

$$\gamma (\mathbf{v})\to e^{-\frac{\theta }{2}{\gamma }_j{\gamma }_k}\gamma (\mathbf{v})e^{\frac{\theta }{2}{\gamma }_j{\gamma }_k}=\gamma (e^{\theta {ϵ}_{jk}}\mathbf{v})$$

the group version. Just as in the symplectic case, exponentiating the $U_L^{\prime }$ only gives a representation up to sign, and one needs to go to the double cover of $SO(n)$ to get a true representation. As in that case, the necessity of the double cover is best seen by use of a complex structure and an analog of the Bargmann-Fock construction, an example will be given in section 31.4.

In order to have a full construction of a quantization of a pseudo-classical system, we need to construct the ${\Gamma }^{+}({\xi }_j)$ as linear operators on a state space. $\mathrm{As}$ mentioned in section 28.2, it can be shown that the real Cliford algebras $\mathrm{Cliff}(r,s,\mathbf{R})$ are isomorphic to either one or two copies of the matrix algebras $M(2^l,\mathbf{R}),\dot{M}(2^l,\mathbf{C})$ , or $M(2^l,\mathbf{H})$ , with the power l depending on $r,s.$ The irreducible representations of such a matrix algebra are just the column vectors of dimension $2^l$ , and there will be either one or two such irreducible representations for $\mathrm{Cliff}(r,s,\mathbf{R})$ depending on the number of copies of the matrix algebra. This is the fermionic analog of the Stone-von Neumann uniqueness result in the bosonic case.

31.1.1 Quantization of the pseudo-classical spin

As an example, one can consider the quantization of the pseudo-classical spin degree of freedom of section 30.3.1. In that case ${\Gamma }^{+}$ takes values in $\mathrm{Cliff}(3,0,\mathbf{R})$ 2 for which an explicit identification with the algebra $M(2,\mathbf{C})$ of two by two complex matrices was given in section 28.2. One has

$${\Gamma }^{+}({\xi }_j)=\frac{1}{\sqrt{2}}{\gamma }_j=\frac{1}{\sqrt{2}}{\sigma }_j$$

and the Hamiltonian operator is

$$\begin{array}{rl}-iH& ={\Gamma }^{+}(h)={\Gamma }^{+}(B_{12}{\xi }_1{\xi }_2+B_{13}{\xi }_1{\xi }_3+B_{23}{\xi }_2{\xi }_3)\\ & =\frac{1}{2}(B_{12}{\sigma }_1{\sigma }_2+B_{13}{\sigma }_1{\sigma }_3+B_{23}{\sigma }_2{\sigma }_3)\\ & =i\frac{1}{2}(B_1{\sigma }_1+B_2{\sigma }_2+B_3{\sigma }_3)\end{array}$$

This is nothing but our old example from chapter $7$ of a fixed spin particle in a magnetic field.

The pseudo-classical equation of motion

$$\frac{d}{dt}{\xi }_j(t)=-\{h,{\xi }_j{\}}_{+}$$

after quantization becomes the Heisenberg picture equation of motion for the spin operators (see equation 7.3)

$$\frac{d}{dt}{\mathbf{S}}_H(t)=-i[{\mathbf{S}}_H\cdot \mathbf{B},{\mathbf{S}}_H]$$

for the case of Hamiltonian

$$H=-\mathbf{\mu }\cdot \mathbf{B}$$

(see equation 7.2) and magnetic moment operator

$$\mathbf{\mu }=\mathbf{S}$$

Here the state space is $\mathcal{H}={\mathbf{C}}^2$ , with an explicit choice of basis given by our chosen identification of $\mathrm{Cliff}(3,0,\mathbf{R})$ with two by two complex matrices. In the next sections we will consider the case of an even dimensional fermionic phase space, but there provide a basis-independent construction of the state space and the action of the Cliford algebra on it.

Chapter contents · Next section: 31.2 The Schr¨odinger representation for fermions: ghosts · 中文版本

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