30.3 Examples of pseudo-classical mechanics

30.3 Examples of pseudo-classical mechanics

In pseudo-classical mechanics, the dynamics will be determined by choosing a Hamiltonian $h$ in ${\mathbf{\Lambda }}^{\ast }({\mathbf{R}}^n)$ . Observables will be other functions $F\in {\Lambda }^{\ast }({\mathbf{R}}^n)$ and they will satisfy the analog of Hamilton’s equations

$$\frac{d}{dt}F=\{F,h{\}}_{+}$$

We’ll consider two of the simplest possible examples.

30.3.1 The pseudo-classical spin degree of freedom

Using pseudo-classical mechanics, a “classical” analog can be found for something that is quintessentially quantum: the degree of freedom that appears in the qubit or spin $\frac{1}{2}$ system that first appeared in chapter 3. Taking $V={\mathbf{R}}^3$ with the standard inner product as fermionic phase space, we have three generators ${\xi }_1,{\xi }_2,{\xi }_3\in V^{\ast }$ satisfying the relations

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

and an 8 dimensional space of functions with basis

$$1,{\xi }_1,{\xi }_2,{\xi }_3,{\xi }_1{\xi }_2,{\xi }_1{\xi }_3,{\xi }_2{\xi }_3,{\xi }_1{\xi }_2{\xi }_3$$

If we want the Hamiltonian function to be non-trivial and of even degree, it will have to be a linear combination

$$h=B_{12}{\xi }_1{\xi }_2+B_{13}{\xi }_1{\xi }_3+B_{23}{\xi }_2{\xi }_3$$

for some constants $B_{12},B_{13},B_{23}$ . This can be written

$$h=\frac{1}{2}\sum _{j,k=1}^3{L}_{jk}{\xi }_j{\xi }_k$$

where $L_{jk}$ are the entries of the matrix

$$L=\left(\begin{array}{ccc}0& B_{12}& B_{13}\\ -B_{12}& 0& B_{23}\\ -B_{13}& -B_{23}& 0\end{array}\right)$$

The equations of motion on generators will be

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

which, since $L=-L^T$ , by theorem 30.1 can be written

$$\frac{d}{dt}{\xi }_j(t)=L{\xi }_j(t)$$

with solution

$${\xi }_j(t)=e^{tL}{\xi }_j(0)$$

This will be a time-dependent rotation of the ${\xi }_j$ in the plane perpendicular to

$$\mathbf{B}=(B_{23},-B_{13},B_{12})$$

at a constant speed proportional to $|\mathbf{B}|$

30.3.2 The pseudo-classical fermionic oscillator

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We have already studied the fermionic oscillator as a quantum system (in section 27.2), and one can ask whether there is a corresponding pseudo-classical system. For the case of d oscillators, such a system is given by taking an even dimensional fermionic phase space $V={\mathbf{R}}^{2d}$ , with a basis of coordinate functions ${\xi }_1,\cdots ,{\xi }_{2d}$ that generate ${\Lambda }^{\ast }({\mathbf{R}}^{2d})$ On generators the fermionic Poisson bracket relations come from the standard choice of positive definite symmetric bilinear form

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

As shown in theorem 30.1, quadratic products ${\xi }_j{\xi }_k$ act on the generators by infinitesimal rotations in the jk plane, and satisfy the commutation relations of so(2d).

To get a pseudo-classical system corresponding to the fermionic oscillator one makes the choice

$$h=\frac{1}{2}\sum _{j=1}^d({\xi }_{2j}{\xi }_{2j-1}-{\xi }_{2j-1}{\xi }_{2j})=\sum _{j=1}^d{\xi }_{2j}{\xi }_{2j-1}$$

This makes h the moment map for a simultaneous rotation in the $2j-1,2j$ planes, corresponding to a matrix in so(2d) given by

$$L=\sum _{j=1}^d{ϵ}_{2j-1,2j}$$

As in the bosonic case, we can make the standard choice of complex structure $J=J_0$ on ${\mathbf{R}}^{2d}$ and get a decomposition

$$V^{\ast }\otimes \mathbf{C}={\mathbf{R}}^{2d}\otimes \mathbf{C}={\mathbf{C}}^d\oplus {\mathbf{C}}^d$$

into eigenspaces of J of eigenvalue $\pm i.$ . This is done by defining

$${\theta }_j=\frac{1}{\sqrt{2}}({\xi }_{2j-1}-i{\xi }_{2j}),{\overline{\theta }}_j=\frac{1}{\sqrt{2}}({\xi }_{2j-1}+i{\xi }_{2j})$$

for $j=1,\dots ,d.$ These satisfy the fermionic Poisson bracket relations

$$\{{\theta }_j,{\theta }_k{\}}_{+}=\{{\overline{\theta }}_j,{\overline{\theta }}_k{\}}_{+}=0,\{{\overline{\theta }}_j,{\theta }_k{\}}_{+}={\delta }_{jk}$$

(where we have extended the inner product $\{\cdot ,\cdot {\}}_{+}$ to $V^{\ast }\otimes \mathbf{C}$ by complex linearity).

In terms of the ${\theta }_j$ , the Hamiltonian is

$$h=-\frac{i}{2}\sum _{j=1}^d({\theta }_j{\overline{\theta }}_j-{\overline{\theta }}_j{\theta }_j)=-i\sum _{j=1}^d{\theta }_j{\overline{\theta }}_j$$

Using the derivation property of $\{\cdot ,\cdot {\}}_{+}$ one finds

$$\{h,{\theta }_j{\}}_{+}=-i\sum _{k=1}^d({\theta }_k\{{\overline{\theta }}_k,{\theta }_j{\}}_{+}-\{{\theta }_k,{\theta }_j{\}}_{+}{\overline{\theta }}_k)=-i{\theta }_j$$

and, similarly,

$$\{h,{\overline{\theta }}_j{\}}_{+}=i{\overline{\theta }}_j$$

so one sees that $h$ is the generator of $U(1)\subset U(d)$ phase rotations on the variables ${\theta }_j$ . The equations of motion are

$$\frac{d}{dt}{\theta }_j=\{{\theta }_j,h{\}}_{+}=i{\theta }_j,\frac{d}{dt}{\overline{\theta }}_j=\{{\overline{\theta }}_j,h{\}}_{+}=-i{\overline{\theta }}_j$$

with solutions

$${\theta }_j(t)=e^{it}{\theta }_j(0),{\overline{\theta }}_j(t)=e^{-it}{\overline{\theta }}_j(0)$$

Previous section: 30.2 Pseudo-classical mechanics and the fermionic Poisson bracket · Next section: 30.4 For further reading · 中文版本

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