# Can we apply superposition principle for states in Fock space?

Please teach me this:
In QTF theory book,I have not seen what saying about probability of creating or anihilating particle(quantum).So I wonder whether we can apply superposition principle for the states with different numbers of particles in Fock space.
Thank you in advanced.

## Answers and Replies

DrDu
This depends on the type of particle. For photons it is possible, e.g. the electric field is defined as an expectation value of creation and anihilation operators. It is not possible for particles that carry charges. States with different charge live in different "superselection sectors". The introduction of gauge groups (which have charge as its generator) is equivalent to that statement, as we can change the phase of a superposition of two states with different charge by a gauge transformation to any value we like. Hence these superspositions are equivalent to mixtures as the phase is undefined.

Thank DrDu very much!

A. Neumaier
This depends on the type of particle. For photons it is possible, e.g. the electric field is defined as an expectation value of creation and annihilation operators. It is not possible for particles that carry charges. States with different charge live in different "superselection sectors". The introduction of gauge groups (which have charge as its generator) is equivalent to that statement, as we can change the phase of a superposition of two states with different charge by a gauge transformation to any value we like. Hence these superpositions are equivalent to mixtures as the phase is undefined.
The last sentence is somewhat inaccurate. What is meant is that it is meaningless to _form_ a superposition in a gauge invariant way, whereas the concept of a mixture makes gauge invariant sense.

Considering a statistic ensemble of states,if we could make a superposition state of this ensemble,then calculating expectative value of something in the superposition state we must account also the off-diagonal elements of ''density matrix''.But in all statistic mechanics(quantum) books they ignore the off-diagonal elements,only considering Trace(density matrix).Please help me this problem.

Are there the ''thisness''(meaning the distinguish) of the states of ensemble in statistic mechanics, so in some sense there is a ''mixture'' in states of ensemble,the states are independend with each other,then we need only the trace?But in QTF theory(high energy) there is not the ''thisness'',the states in ensemble would be transform to each other,so we must account the off-diagonal,according with superposition state of ensemble?

DrDu
Forming the trace of rho A (with A being an arbitrary operator) does not mean that the non-diagonal elements of rho wouldn't contribute.

A. Neumaier
Considering a statistic ensemble of states,if we could make a superposition state of this ensemble,then calculating expectative value of something in the superposition state we must account also the off-diagonal elements of ''density matrix''.But in all statistic mechanics(quantum) books they ignore the off-diagonal elements,only considering Trace(density matrix).Please help me this problem.
Nobody ignores the off-diagonal entires.

The density matrix rho determines expectations of observables A via <A>=Tr rho A.
Here the off-diagonal entries matter. But for A=1 (the identity), you get <A>= Tr rho =1.

Please tell me what is the ''rho''.

DrDu
Please tell me what is the ''rho''.

Sorry,rho is the density matrix.

What does the ''rho'' abbreviate for?

Thanks very much

Can we say in mixture state of a states ensemble,all the states are independent with each other?It seem to me the mixture state is a state that is not able to be described by wave function?

DrDu
Can we say in mixture state of a states ensemble,all the states are independent with each other?It seem to me the mixture state is a state that is not able to be described by wave function?

In ordinary qm I would agree. But I think this depends on the Hilbert space on which you base the theory. E.g. there exists something called "thermo field theory":
http://www.physics.thetangentbundle.net/wiki/Quantum_field_theory/thermo_field_dynamics [Broken]

Also the GNS construction works independently whether the state is pure or not.
Arnold?

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So, what is a general definition of mixed state?Can we use ''entropy'' to describe a mixed state of an ensemble of states?

A. Neumaier
So, what is a general definition of mixed state?Can we use ''entropy'' to describe a mixed state of an ensemble of states?

A state is pure when it cannot be written as a convex combination of other states.
In a fixed Hilbert space, this means that the pure states are just the density matrices of rank 1. (Things are more complicated if there are superselection sectors.)

A. Neumaier
In ordinary qm I would agree. But I think this depends on the Hilbert space on which you base the theory. E.g. there exists something called "thermo field theory":
http://www.physics.thetangentbundle.net/wiki/Quantum_field_theory/thermo_field_dynamics [Broken]

Also the GNS construction works independently whether the state is pure or not.
Arnold?

In general, a state is a positive linear functional from the algebra of observables to the complex numbers, defining the expectation of each bounded observable (and some unbounded ones). The GNS construction produces for each state a corresponding Hilbert space representing the observable algebra. If there are no superselection sectors, all these representations are equivalent, and states can be described by density matrices in any of these Hilbert spaces.

But in statistical mechanics (through the thermodynamic limit) and in relativistic quantum field theory (because of Haag's theorem), there are always superselection sectors, and the density matrix approach is inadequate. (One needs limits of finite volume density matrices.)

The interpretation in terms of ensembles is independent of whether a state is pure or mixed. Even pure states are (in the minimal interpretation of QM) applicable only to ensembles of identically prepared systems. (But of course, the minimal interpretation is incomplete, as we nowadays apply QM routinely to single systems.)

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I have gone through two volumes of Cohen (Quantum Mechanics). Now I need a a book on Quantum Statistical Mechanics. The book should use axiomatic formulation using density matrix. Please give me a suggestion. (I'm also introduced with 2nd quantization slightly)

I have gone through two volumes of Cohen (Quantum Mechanics). Now I need a a book on Quantum Statistical Mechanics. The book should use axiomatic formulation using density matrix. Please give me a suggestion.

A. Neumaier