31.2 The Schr¨odinger representation for fermions: ghosts

31.2 The Schr¨odinger representation for fermions: ghosts

Concept links · terms present in this machine draft; source roles are unverified: irreducible representation · complexification

We would like to construct representations of $\mathrm{Cliff}(r,s,\mathbf{R})$ and thus fermionic state spaces by using analogous constructions to the Schr¨odinger and Bargmann-Fock ones in the bosonic case. The Schr¨odinger construction took the state space $\mathcal{H}$ to be a space of functions on a subspace of the classical phase space which had the property that the basis coordinate functions Poisson-commuted. Two examples of this are the position coordinates $q_j,$ , since $\{q_j,q_k\}=0$ , or the momentum coordinates $p_j.$ since $\{p_j,p_k\}=0$ . Unfortunately, for symmetric bilinear forms $(\cdot ,\cdot )$ of definite sign, such as the positive definite case $\mathrm{Cliff}(n,\mathbf{R})$ 2 the only subspace the bilinear form is zero on is the zero subspace.

To get an analog of the bosonic situation, one needs to take the case of signature $(d,d)$ . The fermionic phase space will then be 2d dimensional, with $d$ dimensional subspaces on which $(\cdot ,\cdot )$ and thus the fermionic Poisson bracket is zero. Quantization will give the Cliford algebra

$$\mathrm{Cliff}(d,d,\mathbf{R})=M(2^d,\mathbf{R})$$

which has just one irreducible representation, ${\mathbf{R}}^{2^d}$ . This can be complexified to get a complex state space

$${\mathcal{H}}_F={\mathbf{C}}^{2^d}$$

This state space will come with a representation of $Spin(d,d,\mathbf{R})$ from exponentiating quadratic combinations of the generators of $\mathrm{Cliff}(d,d,\mathbf{R})$ . However, this is a non-compact group, and one can show that on general grounds it cannot have faithful unitary finite dimensional representations, so there must be a problem with unitarity.

To see what happens explicitly, consider the simplest case $d=1$ of one degree of freedom. In the bosonic case the classical phase space is ${\mathbf{R}}^2$ , and quantization gives operators $Q,P$ which in the Schr¨odinger representation act on functions of $q,$ with $Q=q$ and $P=-i\frac{\partial }{\partial q}$ . In the fermionic case with signature $(1,1)$ basis coordinate functions on phase space are ${\xi }_1,{\xi }_2$ , with

$$\{{\xi }_1,{\xi }_1{\}}_{+}=1,\{{\xi }_2,{\xi }_2{\}}_{+}=-1,\{{\xi }_1,{\xi }_2{\}}_{+}=0$$

Defining

$$\eta =\frac{1}{\sqrt{2}}({\xi }_1+{\xi }_2),\pi =\frac{1}{\sqrt{2}}({\xi }_1-{\xi }_2)$$

one gets objects with fermionic Poisson bracket analogous to those of $q$ and $p$

$$\{\eta ,\eta {\}}_{+}=\{\pi ,\pi {\}}_{+}=0,\{\eta ,\pi {\}}_{+}=1$$

Quantizing, we get analogs of the $Q,P$ operators

$$\hat{\eta }={\Gamma }^{+}(\eta )=\frac{1}{\sqrt{2}}({\Gamma }^{+}({\xi }_1)+{\Gamma }^{+}({\xi }_2)),\hat{\pi }={\Gamma }^{+}(\pi )=\frac{1}{\sqrt{2}}({\Gamma }^{+}({\xi }_1)-{\Gamma }^{+}({\xi }_2))$$

which satisfy anticommutation relations

$${\hat{\eta }}^2={\hat{\pi }}^2=0,\hat{\eta }\hat{\pi }+\hat{\pi }\hat{\eta }=1$$

and can be realized as operators on the space of functions of one fermionic variable $\eta$ as

$$\hat{\eta }=\mathrm{multiplicationby}\eta ,\hat{\pi }=\frac{\partial }{\partial \eta }$$

This state space is two complex dimensional, with an arbitrary state

$$f(\eta )=c_{1}1+c_2\eta$$

with $c_j$ complex numbers. The inner product on this space is given by the fermionic integral

$$(f_1(\eta ),f_2(\eta ))=\int f_1^{\ast }(\eta )f_2(\eta )d\eta$$

with

$$f^{\ast }(\xi )={\overline{c}}_{1}1+{\overline{c}}_2\eta$$

With respect to this inner product, one has

$$(1,1)=(\eta ,\eta )=0,(1,\eta )=(\eta ,1)=1$$

This inner product is indefinite and can take on negative values, since

$$(1-\eta ,1-\eta )=-2$$

Having such negative-norm states ruins any standard interpretation of this as a physical system, since this negative number is supposed to the probability of finding the system in this state. Such quantum systems are called “ghosts”, and do have applications in the description of various quantum systems, but only when a mechanism exists for the negative-norm states to cancel or otherwise be removed from the physical state space of the theory.

Previous section: 31.1 Quantization of pseudo-classical systems · Next section: 31.3 Spinors and the Bargmann-Fock construction · 中文版本

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原书 PDF · 印刷页 335、336、337、338、339、340、341、342、343、344、345、346、347、348

来源版本：2025-10-20

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