Can virtual particles break the law that energy cannot be created or destroyed?

[tom.stoer]

... energy is really not even being "borrowed", but simply transferred. In either case, total system energy is not increased or decreased.

Exactly!

If you look at Feynman diagrams (which are the key ingredient when talking about virtual particles - I will come back to that in a final comment) every theory like QED, QCD, MSSM, SUGRA, ... has a uniquely defined set of rules, a set of "elementary Feynman graphs", namely external particle lines, internal particle lines = propagators and vertices.

By combining these elemenatry graphs you can construct arbitrarily complex graphs. The (infinite) set of all graphs is equivalent to the full perturbation theory of the quantum field theory (I do not say "full theory" as it's only a perturbative treatment which misses certain features, so-called non-perturbarive effects which we know are relevant e.g. in QCD for chiral symmetry breaking and confinement).

Basically the Feynman diagrams are a method of book-keeping and a starting point for calculations. But in popular books they are presented w/o the corresponding mathematical rules (where these come from and how the results are calculated). So let's do the same here.

If you only look at the Feynman graphs diagrammatically you see that different types of particles are converted into each other. One can also equip each line with a "label" for the flow of energy and momentum. (just like an electric current in a wire). At each vertex the flow of energy and momentum is conserved; so along with the particles interacting and changing there is a flow which is conserved meaning that energy and momentum is transferred between the particles at the verices.

One final comment regarding virtual particles: it is not the case that virtual particles do exist in nature and Feynman invented his diagrams in order to describe them. It is more or less the case that virtual particles are "created" by drawing Feynman diagrams. If one would be able to do all the math w/o these diagrams directly and w/o any approximation (perturbation expansion) nobody wouldcare about virtual particles.

 

[sheaf]

One final comment regarding virtual particles: it is not the case that virtual particles do exist in nature and Feynman invented his diagrams in order to describe them. It is more or less the case that virtual particles are "created" by drawing Feynman diagrams. If one would be able to do all the math w/o these diagrams directly and w/o any approximation (perturbation expansion) nobody wouldcare about virtual particles.

I've seen different opinions expressed on the "reality" of virtual particles. I've always argued, as you say, on the side that the particles don't exist - they're only intermediate calculation artifacts. The way I've always tried to explain it is that the diagrams they appear in are a bit like terms in a Taylor series - the full sum of the series is what's important, not the individual terms.

In order to make this argument stronger, if I could argue that the perturbation expansion is not unique, then it would be much harder for people to put the counter argument that the individual terms represent physically real processes. So, my question is (sorry if this is a bit dumb - I'm a novice at this): Is it possible to perform alternative perturbation expansions of an amplitude, that are in some way term by term inequivalent, but sum to the same answer ?

 

[john88888]

one of the physicist I asked said yes it does violate that energy cannot be created or destroyed within the limits of Heisenberg uncertainty principle I assume this is just crap I just want to confirm sorry this is a little over my head but I just curious

 

[tom.stoer]

Is it possible to perform alternative perturbation expansions of an amplitude, that are in some way term by term inequivalent, but sum to the same answer ?

Not really (afaik).

But this is comparable to a Taylor series: the theory of holomorphic functions tells you that the series (that means every term in the series) is uniquely defined, provided the series exists at all and provided you consider the same point x for the expansion.

But based on Taylor series we can consider examples of functions that help to understand the limitations. Consider e.g.

f(x) = e^{-1/x^2}

For this function we known that the function itself as well as all its derivaties satisfy

f^{(n)}(0) = 0

That means that when constructing the Taylor series we get a sum of zeros. That seems to mean that the radius of convergence is exactly zero. Looking at the same function in terms of a complex variable z=x+iy and investigating

f(x=0, y) = e^{1/y^2}

we see that now the function diverges at y=0. So the function does not have a single pole of finite order but a more complicated singularity.

Now let's come back to perturbation theory. What can happen:

• we can start from different vacua resulting in different concepts what a "virtual particle" is.
• there can be non-perturbative effects (e.g. solitons, instantons, ...) which contain something like 1/g (g: coupling constant) for which a Taylor series at g=0 is not well-defined; or which are not visible in perturbation theory because they are eliminated by the approximation.
• there are regimes which cannot be described perturbatively, e.g. confinement in QCD; we know it's there, we can calculate it in lattice gauge theory, but we know for sure that we cannot describe it perturbatively; in this case it's not the concept of virtual particles that ceases to exist, but the concept of real particles in the asymptotic states (plane waves); they do not exist because of confinement.
• we can have a non-trivial fixpoint in the renormalization group flow; in QCD we know that it has a Gaussian fixed point g=0 which means that the theory is asymptotically free; this is the reason why perturbation theory makes sense in QCD; that need not be the case in other theories, which means that g=0 may even be an unphysical point and expansions at g=0 are nonsense.
• we can have a perturbation series which is (after renormalization) finite order-by-order, but which diverges when summing up all terms; btw.: this seems to be the normal behaviour of perturbation series, even for well-established theories like QED and QCD.
• we can have a theory which has some sort of duality; e.g. in two dimensions one can show that certain fermionic theories can be mapped one-to-one to bosonic theories (bosonization, Schwinger model) which are strictly identical in the sense that for an amplitude there are two (numerically identical) expressions, one containing only bosons, one containing only fermions; so what is the particle content of this theory? bosons or fermions?

 

[sheaf]

• we can start from different vacua resulting in different concepts what a "virtual particle" is.
• there can be non-perturbative effects (e.g. solitons, instantons, ...) which contain something like 1/g (g: coupling constant) for which a Taylor series at g=0 is not well-defined; or which are not visible in perturbation theory because they are eliminated by the approximation.
• there are regimes which cannot be described perturbatively, e.g. confinement in QCD; we know it's there, we can calculate it in lattice gauge theory, but we know for sure that we cannot describe it perturbatively; in this case it's not the concept of virtual particles that ceases to exist, but the concept of real particles in the asymptotic states (plane waves); they do not exist because of confinement.
• we can have a non-trivial fixpoint in the renormalization group flow; in QCD we know that it has a Gaussian fixed point g=0 which means that the theory is asymptotically free; this is the reason why perturbation theory makes sense in QCD; that need not be the case in other theories, which means that g=0 may even be an unphysical point and expansions at g=0 are nonsense.
• we can have a perturbation series which is (after renormalization) finite order-by-order, but which diverges when summing up all terms; btw.: this seems to be the normal behaviour of perturbation series, even for well-established theories like QED and QCD.
• we can have a theory which has some sort of duality; e.g. in two dimensions one can show that certain fermionic theories can be mapped one-to-one to bosonic theories (bosonization, Schwinger model) which are strictly identical in the sense that for an amplitude there are two (numerically identical) expressions, one containing only bosons, one containing only fermions; so what is the particle content of this theory? bosons or fermions?

Thanks very much Tom, there are some very persuasive arguments in that list !

 

[Haelfix]

"Not really (afaik)."

Well, there is a trivial way. Namely by changing regularization and renormalization schemes.

Both series could sum to infinity but *naively* differ term by term. Changing renormalization schemes will typically rearrange the kth and (k+1) terms. So they are in effect equivalent, but it might look at first glance like they are not.

 

[tom.stoer]

Good point! One should add this to the list I compiled