30.2 Pseudo-classical mechanics and the fermionic Poisson bracket

30.2 Pseudo-classical mechanics and the fermionic Poisson bracket

Concept links · terms present in this machine draft; source roles are unverified: vector space · linear map · Lie algebra · Lie bracket

The basic structure of Hamiltonian classical mechanics depends on an even dimensional phase space $M={\mathbf{R}}^{2d}$ with a Poisson bracket $\{\cdot ,\cdot \}$ on functions on this space. Time evolution of a function $f$ on phase space is determined by

$$\frac{d}{dt}f=\{f,h\}$$

for some Hamiltonian function $h$. This says that taking the derivative of any function in the direction of the velocity vector of a classical trajectory is the linear map

$$f\to \{f,h\}$$

on functions. As we saw in chapter 14, since this linear map is a derivative, the Poisson bracket will have the derivation property, satisfying the Leibniz rule

$$\{f_1,f_2{f}_3\}=f_2\{f_1,f_3\}+\{f_1,f_2\}{f}_3$$

for arbitrary functions $f_1,f_2,f_3$ on phase space. Using the Leibniz rule and antisymmetry, Poisson brackets can be calculated for any polynomials, just from knowing the Poisson bracket on generators $q_j,p_j(\mathrm{or}$ , equivalently, the antisymmetric bilinear form $\Omega (\cdot ,\cdot ))$ , which we chose to be

$$\{q_j,q_k\}=\{p_j,p_k\}=0,\{q_j,p_k\}=-\{p_k,q_j\}={\delta }_{jk}$$

Notice that we have a symmetric multiplication on generators, while the Poisson bracket is antisymmetric.

To get pseudo-classical mechanics, we think of the Grassmann algebra ${\Lambda }^{\ast }({\mathbf{R}}^n)$ as our algebra of classical observables, an algebra we can think of as functions on a “fermionic” phase space $V={\mathbf{R}}^n$ (note that in the fermionic case, the phase space does not need to be even dimensional). We want to find an appropriate notion of fermionic Poisson bracket operation on this algebra, and it turns out that this can be done. While the standard Poisson bracket is an antisymmetric bilinear form $\Omega (\cdot ,\cdot )$ on linear functions, the fermionic Poisson bracket will be based on a choice of symmetric bilinear form on linear functions, equivalently, a notion of inner product $(\cdot ,\cdot )$

Denoting the fermionic Poisson bracket by $\{\cdot ,\cdot {\}}_{+}$ , for a multiplication anticommutative on generators one has to adjust signs in the Leibniz rule, and the derivation property analogous to the derivation property of the usual Poisson bracket is, for monomials $F_1,F_2,F_3$ 2

$$\{F_1{F}_2,F_3{\}}_{+}=F_1\{F_2,F_3{\}}_{+}+(-1{)}^{|F_2||F_3|}\{F_1,F_3{\}}_{+}{F}_2$$

where $|F_2|$ and $\left|F_3\right|$ are the degrees of $F_2$ and $F_3$ . It will also have the symmetry property

$$\{F_1,F_2{\}}_{+}=-(-1{)}^{|F_1||F_2|}\{F_2,F_1{\}}_{+}$$

and these properties can be used to compute the fermionic Poisson bracket for arbitrary functions in terms of the relations for generators.

The ${\xi }_j$ can be thought of as the “anticommuting coordinate functions” with respect to a basis ${\mathbf{e}}_j$ of $V={\mathbf{R}}^n$ . We have seen that the symmetric bilinear forms on ${\mathbf{R}}^n$ are classified by a choice of positive signs for some basis vectors, negative signs for the others. So, on generators ${\xi }_j$ one can choose

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

with a plus sign for $j=k=1,\cdot \cdot \cdot ,r$ and a minus sign for $j=k=r+1,\cdots ,n$ corresponding to the possible inequivalent choices of non-degenerate symmetric bilinear forms.

Taking the case of a positive-definite inner product for simplicity, one can calculate explicitly the fermionic Poisson brackets for linear and quadratic combinations of the generators. One finds

$$\{{\xi }_j{\xi }_k,{\xi }_l{\}}_{+}={\xi }_j\{{\xi }_k,{\xi }_l{\}}_{+}-\{{\xi }_j,{\xi }_l{\}}_{+}{\xi }_k={\delta }_{kl}{\xi }_j-{\delta }_{jl}{\xi }_k\text{\hspace{1em}(30.1)}$$

and

$$\begin{array}{rl}\{{\xi }_j{\xi }_k,{\xi }_l{\xi }_m{\}}_{+}& =\{{\xi }_j{\xi }_k,{\xi }_l{\}}_{+}{\xi }_m+{\xi }_l\{{\xi }_j{\xi }_k,{\xi }_m{\}}_{+}\\ & ={\delta }_{kl}{\xi }_j{\xi }_m-{\delta }_{jl}{\xi }_k{\xi }_m+{\delta }_{km}{\xi }_l{\xi }_j-{\delta }_{jm}{\xi }_l{\xi }_k\end{array}\text{\hspace{1em}(30.2)}$$

The second of these equations shows that the quadratic combinations of the generators ${\xi }_j$ satisfy the relations of the Lie algebra of the group of rotations in $n$ dimensions $(\mathfrak{so}(n)=\mathfrak{spin}(n))$ . The first shows that the ${\xi }_k{\xi }_l$ acts on the ${\xi }_j$ as infinitesimal rotations in the kl plane.

In the case of the conventional Poisson bracket, the antisymmetry of the bracket and the fact that it satisfies the Jacobi identity imply that it is a Lie bracket determining a Lie algebra (the infinite dimensional Lie algebra of functions on a phase space ${\mathbf{R}}^{2d})$ . The fermionic Poisson bracket provides an example of something called a Lie superalgebra. These can be defined for vector spaces with some usual and some fermionic coordinates:

Definition (Lie superalgebra)

A Lie superalgebra structure on a real or complex vector space $V$ is given by a Lie superbracket $[\cdot ,\cdot {]}_{\pm }$ . This is a bilinear map on $V$ which on generators $X,Y,Z$ (which may be usual or fermionic ones) satisfies

$$[X,Y{]}_{\pm }=-(-1{)}^{|X||Y|}[Y,X{]}_{\pm }$$

and a super-Jacobi identity

$$[X,[Y,Z{]}_{\pm }{]}_{\pm }=[[X,Y{]}_{\pm },Z{]}_{\pm }+(-1{)}^{|X||Y|}[Y,[X,Z{]}_{\pm }{]}_{\pm }$$

where |X| takes value 0 for a usual generator, 1 for a fermionic generator.

Analogously to the bosonic case, on polynomials in generators with order of the polynomial less than or equal to two, the fermionic Poisson bracket $\{\cdot ,\cdot {\}}_{+}$ is a Lie superbracket, giving a Lie superalgebra of dimension $1+n+\frac{1}{2}(n^2-n)$ (since there is one constant, $n$ linear terms ${\xi }_j$ and $\frac{1}{2}\left(n^2-n\right)$ quadratic terms ${\xi }_j{\xi }_k)$ On functions of order two this Lie superalgebra is a Lie algebra, $\mathfrak{so}(n)$ . We will see in chapter 31 that the definition of a representation can be generalized to Lie superalgebras, and quantization will give a distinguished representation of this Lie superalgebra, in a manner quite parallel to that of the Schr¨odinger or Bargmann-Fock constructions of a representation in the bosonic case.

The relation between the quadratic and linear polynomials in the generators is parallel to what happens in the bosonic case. Here we have the fermionic analog of the bosonic theorem 16.2:

Theorem 30.1

The Lie algebra so $(n,\mathbf{R})$ is isomorphic to the Lie algebra ${\Lambda }^2(V^{\ast })$ (with Lie bracket $\{\cdot ,\cdot {\}}_{+})$ of order two anticommuting polynomials on $V={\mathbf{R}}^n$ , by the isomorphism

$$L\leftrightarrow {\mu }_L$$

where $\mathcal{L}\in \mathfrak{so}(n,\mathbf{R})$ is an antisymmetric n by $n$ real matrix, and

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

The $\mathfrak{so}(n,\mathbf{R})$ action on anticommuting coordinate functions is

$$\{{\mu }_L,{\xi }_k{\}}_{+}=\sum _j{L}_{jk}{\xi }_j$$

or

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

Proof

The theorem follows from equations 30.1 and 30.2, or one can proceed by analogy with the proof of theorem 16.2 as follows. First prove the second part of the theorem by computing

$$\begin{array}{rl}{\left\{\frac{1}{2}\sum _{j,k}{\xi }_j{L}_{jk}{\xi }_k,{\xi }_l\right\}}_{+}& =\frac{1}{2}\sum _{j,k}{L}_{jk}({\xi }_j\{{\xi }_k,{\xi }_l{\}}_{+}-\{{\xi }_j,{\xi }_l{\}}_{+}{\xi }_k)\\ & =\frac{1}{2}(\sum _j{L}_{jl}{\xi }_j-\sum _k{L}_{lk}{\xi }_k)\\ & =\sum _j{L}_{jl}{\xi }_j\text{(since L=−LT)}\end{array}$$

For the first part of the theorem, the map

$$L\to {\mu }_L$$

is a vector space isomorphism of the space of antisymmetric matrices and ${\Lambda }^2({\mathbf{R}}^n)$ . To show that it is a Lie algebra isomorphism, one can use an analogous argument to that of the proof of 16.2. Here one considers the action

$$\xi \to \{{\mu }_L,\xi {\}}_{+}$$

of ${\mu }_L\in \mathfrak{so}(n,\mathbf{R})$ on an arbitrary

$$\xi =\sum _j{c}_j{\xi }_j$$

and uses the super-Jacobi identity relating the fermionic Poisson brackets of ${\mu }_L,{\mu }_{L^{\prime }},\xi .$ □

Previous section: 30.1 The Grassmann algebra of polynomials on anticommuting generators · Next section: 30.3 Examples of pseudo-classical mechanics · 中文版本

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