Indefinite Metric and BRST Cohomology

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Author: Dr. Carlo Maria Becchi, Genoa University, Italy
Author: Dr. Camillo Imbimbo, Genoa University, Italy

Indefinite Metric and BRST Cohomology

The standard construction of an indefinite norm space is based on a Hilbert space and on the identification of a metric Hermitian operator J with vanishing kernel. The pseudo inner product in the indefinite norm space is defined by

\langle s|s' \rangle \equiv ( s|J |s' ) \ ,

where the angle brackets define the pseudo inner product in the indefinite norm space while the round ones define the inner product in the Hilbert space. Furthermore the pseudo-adjoint of an operator O is defined by

O^{ \dagger} J=J O^+ \ .

In the B-F oscillator model the total Hilbert space is identified with the Cartesian product of the fermionic and bosonic Fock space. The metric Hermitian operator J is identified using the Pseudo-Hermiticity condition for Q which is equivalent to the Hilbert space relation

Q^{ \dagger} J=J Q\ .

One can solve this relation factorizing J into the product of a bosonic operator j and a fermionic one \mu, finding

J = \mu j=[( \bar a^{ \dagger}-a^{ \dagger})(a- \bar a)+1] \sum_{n=0}^ \infty( \bar A^{ \dagger}-A^{ \dagger})^n(A- \bar A)^n/n!

Given J it is easy to verify that

\bar AJ=J A \ , \ \ AJ=J \bar A \ , \ \ aJ=J \bar a \ , \ \bar aJ=J a \ ,

and hence

\bar A^+= A^{ \dagger} \ , \ \ A^+= \bar A^{ \dagger} \ , \ \ \bar a^+= a^{ \dagger} \ , \ \ a^+= \bar a^{ \dagger} \ .

Furthermore one verifies immediately that both H_{B-F} and Q are Pseudo-Hermitian operators.

Further important points are:

  • The Fock vacuum |0\rangle is an eigenvector of J and has positive pseudo-norm, \langle 0|0 \rangle>0. Furthermore Q|0 \rangle=0.
  • Among the single-particle states, (A^{ \dagger}+ \bar A^{ \dagger})|0 \rangle/ \sqrt{2} and (a^{ \dagger}+ \bar a^{ \dagger})|0 \rangle/ \sqrt{2} have positive pseudo-norm while (A^{ \dagger}- \bar A^{ \dagger})|0 \rangle/ \sqrt{2} and (a^{ \dagger}- \bar a^{ \dagger})|0 \rangle/ \sqrt{2} have negative pseudo-norm.
  • Among the single-particle states, Q \bar A^{ \dagger}|0 \rangle=0 and Q a^{ \dagger}|0 \rangle=0. These relations follow from the nilpotency of Q since \bar A^{ \dagger}|0 \rangle=-iQ \bar a^{ \dagger}|0 \rangle and a^{ \dagger}|0 \rangle=iQA^{ \dagger}|0 \rangle.
  • In general the states of im \ Q are pseudo-orthogonal to those of ker \ Q since Q is pseudo-Hermitian.

To identify ker \ Q with the physical subspace of the indefinite norm Fock space one has to address two issues:

  • Understanding the physical meaning of states pseudo-orthogonal to the rest of ker \ Q such as those in im \ Q.
  • Showing that the states in ker \ Q have non-negative norm.

Regarding the first issue, one observes that adding arbitrary states in im \ Q to states in ker \ Q does not change their pseudo-inner products. Therefore, from the point of view of the physical interpretation based on the probabilistic interpretation of the pseudo-inner product, two states in ker \ Q whose difference belongs to im \ Q must be considered equivalent

|s \rangle \sim \ |t \rangle \Longleftrightarrow |s \rangle - \ |t \rangle \in im \ Q

Hence the linear space of physical states H_{phys} must be identified with ker \ Q/im \ Q, which is the linear space of equivalence classes of vectors in ker \ Q.

As for the second question, if the inner product induced on ker \ Q/im \ Q by the pseudo-inner product on the original space is definite positive, H_{phys} is a Hilbert space. This has to be investigated on a case by case basis.

For the B-F model one proves directly that ker \ Q is the direct sum of im \ Q and the linear span of the vacuum vector |0\rangle. Therefore ker \ Q/im \ Q coincides with the equivalence class of the vacuum |0 \rangle which is a Hilbert space since the pseudo-norm of this state is positive.

Invited by: Dr. Eugene M. Izhikevich, Editor-in-Chief of Scholarpedia, the free peer reviewed encyclopedia
Invited by: Dr. Riccardo Guida, Institut de Physique Théorique; CEA, IPhT; CNRS; Gif-sur-Yvette, France
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