Vacuum in Quantum Field theory is not empty

[emanaly]

I know that the vacuum in Quantum Field theory is not empty, but sometimes I find some people say that the particles are created from nothing because they are created from the vacuum , are those people expression a misleading?

 

[bel]

When we say vacuum, we mean that it is space, emptied of matter. The particles created in a vacuumn are created by pair production when two beams of energetic (in both sense of the word) electromagnetic radiation collide to form a particle- anti-particle pair in accordance with Einstein's energy mass equivalence. In effect, the particles are created out of energy, not exactly nothing.

 

[Norman]

When we say vacuum, we mean that it is space, emptied of matter. The particles created in a vacuumn are created by pair production when two beams of energetic (in both sense of the word) electromagnetic radiation collide to form a particle- anti-particle pair in accordance with Einstein's energy mass equivalence. In effect, the particles are created out of energy, not exactly nothing.

I think the original poster was wondering about pair production in the vacuum with respect to fluctuations due to the Heisenberg Uncertainty Principle. This does not need a reaction to create particles. See for instance the Casimir Effect.

So, the real question is "Can you call the QFT vacuum 'nothing'?"

I don't know. Personally I don't think of it as nothing since I see it as a dynamic system in which these pairs are being created. So in my mind the QFT vacuum is not "nothing."

I think this question may be borderline philosophy however.

 

[emanaly]

I think the original poster was wondering about pair production in the vacuum with respect to fluctuations due to the Heisenberg Uncertainty Principle. This does not need a reaction to create particles. See for instance the Casimir Effect.

So, the real question is "Can you call the QFT vacuum 'nothing'?"

I don't know. Personally I don't think of it as nothing since I see it as a dynamic system in which these pairs are being created. So in my mind the QFT vacuum is not "nothing."

I think this question may be borderline philosophy however.

Yes Norman, you expressed rightly what I mean

 

[humanino]

There are philosophical questions you might ponder about in those matters, somewhere else.
However, in QFT the vacuum is well-defined, if not well-known.
The vacuum in QFT is the state of lowest energy possible.

You can work the mathematics out, and see for yourself that indeed, it is not "empty".
It fluctuates and has non-vanishing average values for some observables.

 

[meopemuk]

I know that the vacuum in Quantum Field theory is not empty, but sometimes I find some people say that the particles are created from nothing because they are created from the vacuum , are those people expression a misleading?

In quantum field theory, the ideas of "vacuum" and "particle" become rather complicated. In the beginning, we define vacuum $|0 \rangle$ as a no-particle state and we define creation operators $a^{\dag}$ which produce 1-particle states by acting on the vacuum

$$|1 \rangle = a^{\dag} |0 \rangle$$

So far, everything is nice and easy. Next, we define the Hamiltonian of our theory, which is an operator expressed as a function of creation and annihilation operators $H(a^{\dag}, a)$.

Now, according to physical intuition, we would expect that our vacuum $|0 \rangle$ is an eigenstate of the Hamiltonian with lowest (zero?) energy. We would also expect one-particle states $|1 \rangle$ to be eigenstates of the Hamiltonian. But this is not true in QFT! It appears that (almost) all Hamiltonians used in QFT do not have $|0 \rangle$ and $|1 \rangle$ as their eigenvectors.

The usual answer to this puzzle is to say that vacuum $|0 \rangle$ is not the real physical vacuum state, and $|1 \rangle$ are not states of real physical particles. It is said that $|0 \rangle$ is a so-called "bare" vacuum, and $|1 \rangle$ are states of "bare" particles. The vacuum and particles we see in reality are called "physical". These physical vacuum $|vac \rangle$ and one-particle $|one \rangle$ states are true eigenvectors of the Hamiltonian, and they are expressed as some complex linear combinations of "bare" particle states. This is the reason why one often hears that vacuum if full of (bare and virtual) particles, and that physical particles are "dressed" by the cloud of (virtual bare) particles.


This situation is a bit paradoxical. In QFT nobody cares about properties of bare particles and states. We want to study physical particles and states. However, our Hamiltonian $H(a^{\dag}, a)$ is expressed through bare particle operators. This makes all calculations and their interpretation very cumbersome.

A great new idea arrived in 1958:

O. W. Greenberg, S. S. Schweber, "Clothed particle operators in simple models of quantum field theory", Nuovo Cim., 8 (1958) 378.

They said (rephrased): "since we don't care about bare particles, there is no reason to keep their creation and annihilation operators in the theory. Let's express our Hamiltonian directly in terms of creation and annihilation operators of "physical " or "dressed" particles. Let's work directly with the full physical vacuum state $|vac \rangle$." Greenberg and Schweber were able to show that "bare" particles can be eliminated and quantum field theories can be reformulated in this "dressed particle" picture without losing anything of importance. In this picture, vacuum $|vac \rangle$ is a no-particle state (there are no "physical" particles in vacuum). One-particle states $|one \rangle$ have just one (physical) particle in them. So, our physical intuition should not be offended.

So, in answering you question about vacuum filled with virtual particles, I should say that this (unfortunately widespread) idea is an artefact of using unphysical (bare) states as our basis in QFT. In fact, there are no "physical" particles in the "physical" vacuum.

Eugene.

 

[Barmecides]

So, in answering you question about vacuum filled with virtual particles, I should say that this (unfortunately widespread) idea is an artefact of using unphysical (bare) states as our basis in QFT. In fact, there are no "physical" particles in the "physical" vacuum.

Eugene.

Thanks a lot meopemuk for this great answer very clear !
By the way, when we are talking about vacuum in electroweak symmetry breaking for example, are we talking about |0> or |vac> ?

 

[meopemuk]

By the way, when we are talking about vacuum in electroweak symmetry breaking for example, are we talking about |0> or |vac> ?

Unfortunately, gauge field theory (including spontaneous symmetry breaking and confinement) is formulated in a language that is quite different from what I presented. I am still struggling to understand its exact meaning. So, I'll let others to answer your question.

Eugene.

 

[Nick666]

Cant it be that the fabric of space-time continuum or space is transforming into the virtual particle and vice-versa?

 

[meopemuk]

Cant it be that the fabric of space-time continuum or space is transforming into the virtual particle and vice-versa?

As I tried to explain, virtual (and bare) particles should be understood as mathematical fictions that are characteristic for one particular (perturbative) formulation of quantum field theory. Virtual particles can be met at certain intermediate steps of calculations (e.g., Feynman diagrams), however, they cannot be directly observed in real life.

I don't know what definition of "the fabric of space-time continuum" you have in mind, but I suspect, it is something as fictitious as virtual particles. Can two unreal things transform into each other? Sure. But this is not question about physics, in my opinion.

Physics is about things that we can observe in experiments: energies of atomic transitions, scattering cross-sections, positions and momenta of real particles, etc. If you want to get a satisfactory answer, you should try to formulate your question in this language.

Eugene.

 

[Nick666]

Do virtual particles have size? Comparable to Planck size. (length)

I heard that when the latest mega particle accelerator will function, they will discover new and exciting things, and new particles, and they might discover that space-time is fabricated of Planck size particles.

 

[olgranpappy]

It fluctuates and has non-vanishing average values for some observables.

Embarrassingly non-vanishing in some cases...

E.g., RMS electric field in the vacuum:

$$<0|\vec E|0>=0$$
Great!

$$<0|E^2|0>=\infty$$
Uh oh!

 

[meopemuk]

Do virtual particles have size? Comparable to Planck size. (length)

No.

I heard that when the latest mega particle accelerator will function, they will discover new and exciting things,

Most certainly, yes.

and new particles,

Possible

and they might discover that space-time is fabricated of Planck size particles.

No way.
Eugene.

 

[lightarrow]

Embarrassingly non-vanishing in some cases...

E.g., RMS electric field in the vacuum:

$$<0|\vec E|0>=0$$
Great!

$$<0|E^2|0>=\infty$$
Uh oh!

Sorry, in which region of space such average values are computed?

 

[olgranpappy]

Sorry, in which region of space such average values are computed?

I'm not averaging with respect to space, I'm evaluating the expectation value of the electric field with respect to the ground state.

If you are asking me what is the spatial argument of the electric field... it doesn't matter, all points of empty space are equivalent.

 

[meopemuk]

Embarrassingly non-vanishing in some cases...

E.g., RMS electric field in the vacuum:

$$<0|\vec E|0>=0$$
Great!

$$<0|E^2|0>=\infty$$
Uh oh!

Hi olgranpappy,

I wouldn't worry about this infinity at all. First, in QED the "electric field" $\vec E$ is a certain combination of photon quantum fields, or equivalently, a certain function of photon creation and annihilation operators. It is questionable whether this combination has anything to do with real macroscopic observable electric fields. At least, I haven't seen reliable (and comparable with experiment) calculations of macroscopic electric fields within QED. The main purpose of QED is to calculate the scattering matrix (S-matrix) for physical particles and some other properties (e.g., energies of bound states) that are directly related to the S-matrix. And QED does that brilliantly.

Second, it is even questionable whether electric fields are truly observable physical objects. When we "measure" the electric field at point x we, actually, place a test charge at point x and measure its acceleration. The acceleration is created, of course, by interactions of this test charge with other charges around it. It can be debated whether this interaction is transmitted by some independent agent (also known as "field", which, presumably, has energy and momentum of its own) or that we are dealing with "action-at-a-distance". Since fields are not measurable directly, it is, at least, conceivable to think that a theory can be formulated in such a form that electric fields (just as virtual and bare particles discussed in my previous posts) can be eliminated from the theory without any adverse effect. The "dressed particle" theory achieves exactly that.

Of course, I am not talking here about free transverse electromagnetic fields associated with propagating light. They are known to be directly measurable and possesses their own momentum and energy. However, within QED these transverse "fields" are better described as collections of large numbers of discrete physical particles - photons. The same description is valid within the "dressed particle" version of QED. And the energy contained in this "field" is just the sum of energies of all photons, so it cannot be infinite. So, there is no contradiction.

Eugene.

 

[olgranpappy]

Hi olgranpappy,

I wouldn't worry about this infinity at all. First, in QED the "electric field" $\vec E$ is a certain combination of photon quantum fields, or equivalently, a certain function of photon creation and annihilation operators. It is questionable whether this combination has anything to do with real macroscopic observable electric fields. At least, I haven't seen reliable (and comparable with experiment) calculations of macroscopic electric fields within QED. The main purpose of QED is to calculate the scattering matrix (S-matrix) for physical particles and some other properties (e.g., energies of bound states) that are directly related to the S-matrix. And QED does that brilliantly.

Oh yes, I agree with you. I just find the example I gave amusing.

 

[Nick666]

Why don't virtual particles have size(length)?

Is their size (length) zero ?

Why can't they discover that space is composed of particles of Planck size (length) or much smaller. or whatever particles of whatever size?

 

[olgranpappy]

Why don't virtual particles have size(length)?

I hate to say it, but as usual, it gets down to what exactly do you mean by "size(length)?"

But regardless, you can't just go up with a ruler and measure a virtual particle because they only exist as "intermediate states" and not as what actually comes out of, say, a scattering experiment.

Is their size (length) zero ?

no.

Why can't they discover that space is composed of particles of Planck size (length) or much smaller. or whatever particles of whatever size?

Look. That just doesn't make sense. And besides, it's seems to be just a random statement that you plucked out of thin air.

 

[Nick666]

Then why do I see physicist saying all the time "the fabric of space-time continuum...bla bla" , "space expands...bla bla" ?

What exactly is expanding?

What is the fabric of space-time ?

What is it composed of?

Is it composed of nothing?

Is it composed of something?

Do I define it as the quantum vacuum (field), the lowest energy state ?

If so, what could that energy transform into ? (from what I know energy and matter can change into one another...d`oh)

 

[meopemuk]

Then why do I see physicist saying all the time "the fabric of space-time continuum...bla bla" , "space expands...bla bla" ?

What exactly is expanding?

What is the fabric of space-time ?

What is it composed of?

Is it composed of nothing?

Is it composed of something?

Do I define it as the quantum vacuum (field), the lowest energy state ?

If so, what could that energy transform into ? (from what I know energy and matter can change into one another...d`oh)

These are very deep questions, and I don't think there exists a consensus opinion among practicing scientists. If you decided to learn general relativity, you would have definitely noticed that space-time there plays a role of a physical object, which has its own energy-momentum and can "interact" with matter. Some modern attempts to quantize gravity also treat space-time as a kind of "fabric", whose physical properties (e.g., dimension, Planck-scale structure) emerge from something even more fundamental (?).

In my personal opinion (which, I should warn you, is not the "mainstream"), all this "space-time continuum" stuff is not relevant to physics. All we need to know is that there exists an observable called "position", and that each measurement can be labeled by the parameter "time", which is simply the reading of the laboratory clock at the instant of the measurement.

No "fabric", no "field", no "energy", no "structure", no bla-bla...

Eugene.

 

[Nick666]

You mean space is ... nothing ?

 

[meopemuk]

You mean space is ... nothing ?

Yes, that's what I mean.

In quantum mechanics there is one place where we meet the "position space" with coordinates (x,y,z). This is the 3-dimensional "space" of common eigenvalues of 3 commuting components of the position operator. Wavefunctions in the position representation are defined on this "space". In quantum mechanics we are free to choose any other set of mutually commuting operators and define our wavefunctions on their common eigenvalues. For example, we can choose 3 components of the momentum operator to define wavefunctions in the momentum space.

There is no any fundamental difference between position-space and momentum-space wavefunctions in quantum mechanics. However, we still have this intuitive belief that the position space is somehow more important and more fundamental. Indeed our organs of sensation (e.g., vision) tell us that we are "embedded" in the 3D position space, not in the 3D momentum space. Why it is so? I think the reason is that by natural selection nature found that it is preferable for our survival to have this "position space" feeling. One can imagine that in other conditions living creatures could develop a primary sensation of the momentum space, and keep the idea of the position space as an afterthought, if it would be benefitial for their survival.

On the other hand, if we assume that the position space is some fundamental "continuum" with its "structure", "curvature", "energy", etc. then we need to explain how this vision corresponds to the QM position space described as common eigenvalues of the position operator, which I mentioned above. I believe that we cannot have two different notions for the same thing (space).

Anyway, these are my ideas about space. I understand that they are controversial, crazy, and perhaps not worth your attention, but that's what I've got.

Eugene.

 

[Nick666]

But how is it that matter can move through space? Or in fact, space "is created" by motion ?

 

[olgranpappy]

But how is it that matter can move through space? Or in fact, space "is created" by motion ?

First of all, this question is ridiculous. Second of all, here is an experiement that you can try at home to convince yourself that matter *does*, in fact, move throught space:

1. Hold your hand in front of your face.

2. Now move it back and forth.

You are now observing matter moving through space. Did the matter move through space? Yes.

Was space "created by motion?" No. In fact, that just doesn't make a lick of sense.

Cheers.

 

[Nick666]

Yeah, but there was matter that moved here even before the Earth was created, let alone my hand. So my hand isn't the one who was in motion first.

I`ll ask another way. Was space created by motion of matter? (lets say the first motion of matter, whatever that may be)

That doesn't matter anyway, but I hope meomepuk will answer.

And I still have trouble with space being nothing, considering some galaxies are drifting apart with speeds >c .

 

[olgranpappy]

Yeah, but there was matter that moved here even before the Earth was created, let alone my hand. So my hand isn't the one who was in motion first.

I`ll ask another way. Was space created by motion of matter? (lets say the first motion of matter, whatever that may be)

no.

That doesn't matter anyway, but I hope meomepuk will answer.

sure. me too.

And I still have trouble with space being nothing, considering some galaxies are drifting apart with speeds >c .

What does that have to do with anything? As you may or may not be aware, relative velocities are not limited to v<c.

 

[meopemuk]

But how is it that matter can move through space? Or in fact, space "is created" by motion ?

If we accept (as I am trying to convince you) that space is nothing more than (expectation) values of the position (operator) of particles of matter, then the answer to your question is simple. Movement of particles is synonymous with the time dependence of (eigenvalues of the) position (operator). The time dependence of the position operator of a particle is given by the usual quantum mechanical formula

$$\mathbf{R}(t) = e^{\frac{i}{\hbar}Ht} \mathbf{R}(0) e^{-\frac{i}{\hbar}Ht}$$

If the particle is non-interacting ($H=H_0$), then this formula yields simply

$$\mathbf{R}(t) = \mathbf{R}(0) + \mathbf{V}t$$

If the particle interacts with other particles, then $H \neq H_0$ and the time evolution is more complicated. The expectation values of position are obtained by taking matrix elements of these operators on corresponding particle states.

In this algebraic approach we don't need to ask philosophical questions like "what is space?", "what is its structure?", "what are its properties?" We obtain quantities (expectation values of position of physical particles of matter) that can be directly compared with experiment. I think this is the main purpose of any good theory.

Eugene.

 

[Nick666]

Is space similar to an empty set in math ?

"In mathematics, nothing does not have a technical meaning. It could be said that a set contains "nothing" if and only if it is the empty set, in which case its cardinality (or size) is zero. In other words, the word "nothing" is an informal term for an empty set." http://en.wikipedia.org/wiki/Nothing#Science

http://en.wikipedia.org/wiki/Empty_set

http://en.wikipedia.org/wiki/Axiom_of_empty_set

 

[meopemuk]

I`ll ask another way. Was space created by motion of matter? (lets say the first motion of matter, whatever that may be)

In my opinion, physics is an experimental science. A correctly formulated physical problem should begin with a description of how the (state of the) physical system is prepared and what property we are measuring. Then the role of the theory is to predict results of these measurements, and how these results change with the change of the observer. If we can make these predictions, and they match with experiment, then we are done.

Your question is not formulated this way, so I am afraid it doesn't have a good answer. Space, by itself, doesn't have any measurable properties. If we place all kinds of measuring devices (bubble chambers, Geiger counters, Stern-Gerlach apparatuses,...) in empty space, they will stay silent. So, empty space (or vacuum) is not a true physical system. One can also say that it is the most trivial physical system that doesn't have any properties. There is nothing interesting for physics in the empty space.

Eugene.

 

[olgranpappy]

Is space similar to an empty set in math ?

No. Space is like $$R^3$$ the set of all ordered triples of real numbers.

 

[Nick666]

And the real numbers corespond to what?

 

[olgranpappy]

points in space.

 

[Nick666]

And the points corespond to ?

 

[meopemuk]

And the points corespond to ?

In my opinion, we need to distinguish two things: "vacuum" and "position space".

Vacuum (or "empty space") is an example of a physical system from which all particles has been removed. It is simply a no-particle physical system. So, vacuum is somewhat analogous to the "empty set"

Position space is a linear 3-dimensional space $R^3$ which is a common set of eigenvalues of three commuting components of the position operator. Points in this space enumerate possible values that can be obtained by measuring the observable called "position". Position-space wavefunctions are defined on this space. This is an abstract space, similar to the momentum space or any other common set of eigenvalues of commuting operators in quantum mechanics.

Eugene.