Perturbative and non perturbative vaccum states.

[dpa]

Hi all,

what is the meaning/difference between perturbative and non perturbative vaccum.

 

[tom.stoer]

Let's start with a simple explanation:

The perturbative or Fock vacuum |0> is simple described by

a_n|0> = 0

for all possible quantum numbers n.


The non-perturbative ground state is describe by something like

(H-E°)|Ω> = 0

with minimum E°.


Alternatively one could write something like

<0|H|0> ≥ <Ω|H|Ω>

 

[dpa]

hi,
could you give physical meaning rather than mathematical.
I am not an expert you see.

 

[lfighter]

A non perturbative vacuum may be topologically different from the trivial or ordinary vacuum. That is, one cannot use a topologically trivial transformation(homotopic to identical mapping) to transform it to the trivial vacuum. You can read some references on instanton to get more information.

 

[tom.stoer]

Typically a non-perturbative vacuum is not 'empty'. In QCD you have chiral symmetry breaking with a non-vanishing order parameter indicating a phase transition. The order parameter is the so-called quark condensate \langle \bar{q}q\rangle \neq 0. That means that in the phase where chiral symmetry is broken the 'vacuum' is not 'empty' but 'contains quark-antiquark pairs'.

Usually you would assume that \langle \bar{q}q\rangle = 0 b/c of normal ordering, but this applies only to the trivial vacuum state.

 

[Naty1]

Hi dpa:
I'm not getting this yet...

"Typically a non-perturbative vacuum is not 'empty'...

is a perturbative vacuum 'empty'...? that doesn't sound like this description:...[...the article provides some interesting background]

Non-vanishing vacuum state

If the quantum field theory can be accurately described through perturbation theory, then the properties of the vacuum are analogous to the properties of the ground state of a quantum mechanical harmonic oscillator (or more accurately, the ground state of a QM problem). In this case the vacuum expectation value (VEV) of any field operator vanishes. For quantum field theories in which perturbation theory breaks down at low energies (for example, Quantum chromodynamics or the BCS theory of superconductivity) field operators may have non-vanishing vacuum expectation values called condensates. In the Standard Model, the non-zero vacuum expectation value of the Higgs field, arising from spontaneous symmetry breaking, is the mechanism by which the other fields in the theory acquire mass.

http://en.wikipedia.org/wiki/Vacuum_state#Non-vanishing_vacuum_stateand this:

...Vacuum energy is the zero-point energy of all the fields in space...the energy of the vacuum, which in quantum field theory is defined not as empty space but as the ground state of the fields...
The zero-point energy is ...the expectation value of the Hamiltonian; here, however, the phrase vacuum expectation value is more commonly used, and the energy is called the vacuum energy...

http://en.wikipedia.org/wiki/Zero-point_energy

but I'm still not clear about the answer to your question...

 

[tom.stoer]

I'm not getting this yet...

"Typically a non-perturbative vacuum is not 'empty'...

is a perturbative vacuum 'empty'...? that doesn't sound like this description ??

Yes, in a certain sense the perturbative vacuum is 'empty'; it's annihilated by typical field operators, so the result for counting particles in the vacuum is zero (after normal ordering); the examples you give (condensates like BCS, QCD ground state, non-vanishing vev for Higgs, ...) are all examples for non-perturbative vacuum states.

 

[Naty1]

...so the result for counting particles in the vacuum is zero

you Are referring to particles, here, not virtual particles...right?

 

[tom.stoer]

yes!

 

[marcus]

Is the number of particles in a region always well-defined?
Say in the case the geometry is curved, or there are different observers?
I've heard people say it's not a well-defined concept.

 

[atyy]

hi,
could you give physical meaning rather than mathematical.
I am not an expert you see.

The non-perturbative vacuum is the true ground state of a system.

If the system is strongly interacting, then we may not know how to solve our equations to get the true ground state. If there is a non-interacting system whose ground state we do know, then we may try to write the true ground state approximately as the ground state of the non-interacting system plus some, hopefully small, corrections. (That doesn't always work.)

 

[martinbn]

Is the number of particles in a region always well-defined?
Say in the case the geometry is curved, or there are different observers?
I've heard people say it's not a well-defined concept.

No, it is not well defined in curved beckground, in the sense that it is coordinate dependent. But i have also heard that even in flat spacetime (and only inertial systems considered) 'particle' is not well defined in a bounded region.

 

[marcus]

Is the number of particles in a region always well-defined?
Say in the case the geometry is curved, or there are different observers?
I've heard people say it's not a well-defined concept.

No, it is not well defined in curved beckground, in the sense that it is coordinate dependent. But i have also heard that even in flat spacetime (and only inertial systems considered) 'particle' is not well defined in a bounded region.

I have seen what you heard demonstrated mathematically*. It seems that the idea of "particle" and the number of particles taking part in any given circumstance is highly observer dependent and geometry dependent.

"Particle" seems very far from being a fundamental, background independent, concept. More of a mathematical convenience useful in specific circumstances. Rather than something in nature.

Vacuum also observer dependent.

*Google "rovelli particle" and get http://arxiv.org/abs/gr-qc/0409054 What is a particle?

 

[Ilmrak]

No, it is not well defined in curved beckground, in the sense that it is coordinate dependent. But i have also heard that even in flat spacetime (and only inertial systems considered) 'particle' is not well defined in a bounded region.

Exactly, even in QFT the number of particles in any bounded region is not an observable.
The particle number operators in 2 bounded disjoint region do not commute for space-like separations (i's not a local function of the fields), they only approximately commute for high separations.

What is an observable is for example the charge contained in any bounded region.

Ilm

 

[martinbn]

I have seen what you heard demonstrated mathematically*. It seems that the idea of "particle" and the number of particles taking part in any given circumstance is highly observer dependent and geometry dependent.

"Particle" seems very far from being a fundamental, background independent, concept. More of a mathematical convenience useful in specific circumstances. Rather than something in nature.

Vacuum also observer dependent.

*Google "rovelli particle" and get http://arxiv.org/abs/gr-qc/0409054 What is a particle?

Yes, I have seen that too. My comment was that even after fixing the background even in flat spacetime QFT, the concept of particle is difficult. Ilmrak's post explains better, what I've only heard.

 

[Naty1]

Starting with just

Is the number of particles in a region always well-defined?
Say in the case the geometry is curved, or there are different observers?
I've heard people say it's not a well-defined concept.

I'd be really doubtful on a conceptual basis that a particle would be well defined if we know all our best theories falter at the singularity of a black hole...are there even 'particles' [mass, space] in those extreme conditions...where do they 'go' ...in fact where does space go...that seems a 'singularity' in time...it only takes a single exception like this to hint the rest of GR and QM are most likely approximations...it's perhaps not just particles that may not be so well defined as we take them to be in everyday less extreme conditions.

String theory suggests that it may be the configuration of higher dimensional spaces that influences string [particles] properties...their vibrational patterns and energies for example ...so when spacetime jiggles around or morphs from one region to another it seems plausible that our perception of particles might also change...because they change.

Further, the Unruh effect [regarding vacuum state temperatures] suggests different observers read coincident spacetime vacuum conditions differently...another hint that things globally and locally may be more surprising then we expect.

Both these concepts seem to support Marcus post.

I read Rovelli's Introduction and this caught my attention:

..uniquely-defined particle states do not exist in general, in QFT on a curved spacetime. ... in general, particle states are difficult to define in a background-independent quantum theory of gravity.

Not surprising...
I put the article on my reading list...

 

[Naty1]

For those who might be interested in a complementary discussion, with some good explanations of the mathematical apparatus involved, check this one from March 2010:

What is a Particle.
https://www.physicsforums.com/showthread.php?t=386051

More on Fock states, operators,Poincaire Group, local versus global representations, etc,etc..it's a long one.


After I started reading 'What is a Particle' this time around I thought 'This sounds familiar' whereupon my enfeebled memory kicked in and I found Marcus had previously baited me on the same topic [!] instigating the 2010 discussion.

[Note: I still don't understand the rules here on posting in old threads, but I recently got 'censored' by a moderator when I inadvertently posted an old one...so be warned!]