Expectation value of Normal ordered Stress-energy tensor

In summary, the authors discuss the normal ordering of operators in terms of free fields and the stress tensor. They show that for operators quadratic in the fields, the normal-ordered expression reduces to a constant c-number, which is the only contribution to the vacuum expectation value. This is analogous to a single harmonic oscillator. The conversation then dives into a discussion about this concept and its application to an object like \langle \psi|:\phi^2:|\psi\rangle.
  • #1
LAHLH
409
1
Hi,

In Birrel and Davies ch4 they write:

[tex] \langle \psi|:T_{ab}:|\psi \rangle =\langle \psi|T_{ab}|\psi \rangle -\langle 0|T_{ab}|0 \rangle [/tex]

this is for the usual Mink field modes and vac state. Why does normal ordering reduce to this expression, could anybody point me the way to deriving this?
 
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  • #2
In terms of free fields, the stress tensor is quadratic, so the difference with the normal-ordered expression is just a (divergent) c-number. This c-number is the only contribution to the vacuum expectation value, so you find the expression in question.

An analogous expression holds for any operator quadratic in the fields, as some playing around with annihilation and creation operators will reveal. You can already see how it works for a single harmonic oscillator if you consider the operator [itex]\hat{\mathcal{O}}= \alpha a^\dagger a + \beta a a^\dagger[/itex] and expectation values in the [itex]|n\rangle[/itex] state.
 
  • #3
fzero said:
In terms of free fields, the stress tensor is quadratic, so the difference with the normal-ordered expression is just a (divergent) c-number. This c-number is the only contribution to the vacuum expectation value, so you find the expression in question.

An analogous expression holds for any operator quadratic in the fields, as some playing around with annihilation and creation operators will reveal. You can already see how it works for a single harmonic oscillator if you consider the operator [itex]\hat{\mathcal{O}}= \alpha a^\dagger a + \beta a a^\dagger[/itex] and expectation values in the [itex]|n\rangle[/itex] state.

Thanks for the reply.

Taking your harmonic oscillator example I think I see what you're getting at:

[itex] \langle n|:\hat{\mathcal{O}}:|n\rangle =\langle n|\alpha a^{\dagger}a+\beta a^{\dagger}a|n\rangle[/itex]

then use [itex]aa^{\dagger}-a^{\dagger}a=1[/itex]

to give:

[itex] \langle n|:\hat{\mathcal{O}}:|n\rangle =\langle n|\alpha a^{\dagger}a+\beta (aa^{\dagger}-1)|n\rangle=\langle n|\hat{\mathcal{O}}|n\rangle-\beta\langle n|n\rangle=\langle n|\hat{\mathcal{O}}|n\rangle-\beta[/itex]

But actuallythe VEV [itex]\langle 0|\hat{\mathcal{O}}|0\rangle =\langle 0|\alpha a^{\dagger}a+\beta aa^{\dagger}|0\rangle=\beta\langle 0| aa^{\dagger}|0\rangle=\beta\langle 1|1\rangle =\beta[/itex]

So we could have wrote

[itex] \langle n|:\hat{\mathcal{O}}:|n\rangle=\langle n|\hat{\mathcal{O}}|n\rangle-\langle 0|\hat{\mathcal{O}}|0\rangle [/itex]

I just need to convince myself this works for an object like [itex]\langle \psi|:\phi^2:|\psi\rangle [/itex] now...
 

1. What is the expectation value of the normal ordered stress-energy tensor?

The expectation value of the normal ordered stress-energy tensor is a quantity that is used to describe the average energy and momentum density of a quantum field. It is calculated by taking the average of the normal ordered stress-energy tensor over all possible quantum states.

2. How is the expectation value of the normal ordered stress-energy tensor calculated?

The expectation value of the normal ordered stress-energy tensor is calculated by first defining the normal ordered stress-energy tensor as the sum of the energy and momentum operators, with the creation and annihilation operators ordered in a specific way. Then, the expectation value is calculated by taking the average of this operator over all possible quantum states.

3. What is the physical significance of the expectation value of the normal ordered stress-energy tensor?

The expectation value of the normal ordered stress-energy tensor is physically significant because it describes the average energy and momentum density of a quantum field. It is used in calculations of various physical processes, such as particle interactions and vacuum energy.

4. How does the expectation value of the normal ordered stress-energy tensor differ from the regular stress-energy tensor?

The expectation value of the normal ordered stress-energy tensor differs from the regular stress-energy tensor because it takes into account the ordering of the creation and annihilation operators. This is important in quantum field theory, where the ordering of operators can affect the results of calculations.

5. What are the applications of the expectation value of the normal ordered stress-energy tensor?

The expectation value of the normal ordered stress-energy tensor has many applications in theoretical physics, particularly in quantum field theory. It is used in calculations of vacuum energy, particle interactions, and the behavior of quantum fields in curved spacetime. It is also used in the study of black holes and cosmology.

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