# Virtual particles in rigorous quantum field theory

#### mattt

"virtual particles" in rigorous quantum field theory

If I am not mistaken "virtual particles" are just a name someone put to some integrals that we use to calculate different things, and those integrals depends on the perturbation scheme and on the gauge selected, and they don't even exist in other computation techniques (Lattice QFT) that we can use (in principle) to answer the very same questions that Perturbation techniques can answer.

Then, when I read things like:

"Weak neutral currents interactions are mediated by the Z boson. The discovery of weak neutral currents was a significant step toward the unification of electromagnetism and the weak force into the electroweak force, and led to the discovery of the W and Z bosons".

I suppose they mean the discovery/confirmation of the W boson field and the Z boson field (but not the existence of the virtual W and virtual Z), because in principle (I guess) we could have done the pertinent computations with Lattices (if we had super-fast computers) or maybe other computation technique and have discovered/explained as well the neutral currents without the use of virtual Z exchanges, is it right?

If I remember corretly, in the most rigorous treatment of QFT (Axiomatic and Constructive Quantum Field Theory) the concept of "virtual particles" does not appear, does not exist. Even the concept of "real particles" is only approximate there ( when the field expectation is non-zero only in a localized small region, isn't it?) and I guess that that (when the field expectation is non-zero only in a localized small region) is the only case when we can properly speak of "a real photon", "a real Z or real W", "a real gluon", whereas we could get rid of the concept of "virtual particles" without changing the mathematical definition of the theory at all.

Last question: When a QFT can be defined with total mathematical rigour (like many QFT in 2 and 3 dimensions, for example QED in fewer dimensions) I think it has been proved that perturbative calculations and non-perturbative calculations (Lattices, others? ) agree, they are equivalent methods of doing mathematical computations (in theory, not necessary in practice). But QED in four dimensions is not under the control of "Rigorous Quantum Field Theory" so...imagine for a moment that tomorrow it is mathematically proved that QED in four dimensions can not verify the Wightman Axioms, that is, it is not mathematically consistent. In this hipothetical case, we could not assure that the perturbative calculations and other techniques (as Lattices) would agree (even with super-fast computers). In this hipothetical scenario, maybe the perturbative calculations are the only ones that leads to facts as Lamb shift and anomalous magnetic moment. But anyway, that
would not change the fact that "virtual particles" are some mathematical integrals in some computations techniques, so I would not see any fundamental change (relative to the "status" of virtual particles) even in this hipothetical scenario.

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#### A. Neumaier

Re: "virtual particles" in rigorous quantum field theory

If I am not mistaken "virtual particles" are just a name someone put to some integrals that we use to calculate different things, and those integrals depends on the perturbation scheme and on the gauge selected, and they don't even exist in other computation techniques (Lattice QFT) that we can use (in principle) to answer the very same questions that Perturbation techniques can answer.
That's a concise summary of what is stated in Chapter A7 of my theoretical physics FAQ at http://www.mat.univie.ac.at/~neum/physfaq/physics-faq.html#A7 .
Then, when I read things like:

"Weak neutral currents interactions are mediated by the Z boson. The discovery of weak neutral currents was a significant step toward the unification of electromagnetism and the weak force into the electroweak force, and led to the discovery of the W and Z bosons".
I suppose they mean the discovery/confirmation of the W boson field and the Z boson field (but not the existence of the virtual W and virtual Z), because in principle (I guess) we could have done the pertinent computations with Lattices (if we had super-fast computers) or maybe other computation technique and have discovered/explained as well the neutral currents without the use of virtual Z exchanges, is it right?
The ''mediation'' refers to the virtual particle imagery for the corresponding interaction terms in the standard model Lagrangian, whereas ''detection'' refers to the detection of real, on-shell W and Z particles.
If I remember correctly, in the most rigorous treatment of QFT (Axiomatic and Constructive Quantum Field Theory) the concept of "virtual particles" does not appear, does not exist.
Indeed, the concept doesn't figure there.
Even the concept of "real particles" is only approximate there ( when the field expectation is non-zero only in a localized small region, isn't it?) and I guess that that (when the field expectation is non-zero only in a localized small region) is the only case when we can properly speak of "a real photon", "a real Z or real W", "a real gluon", whereas we could get rid of the concept of "virtual particles" without changing the mathematical definition of the theory at all.
The conventional usage is ''photon state'' for an arbitrary single excitation of the photon field, localized or not, while a particle picture is semiclassical, requiring at least localization close to a ray. In QED, charged particles cannot be localized better because of gauge invariance.

This is the rigorous use of the language - of course one can find much tolerated abuse
Last question: When a QFT can be defined with total mathematical rigour (like many QFT in 2 and 3 dimensions, for example QED in fewer dimensions) I think it has been proved that perturbative calculations and non-perturbative calculations (Lattices, others? ) agree, they are equivalent methods of doing mathematical computations (in theory, not necessary in practice). But QED in four dimensions is not under the control of "Rigorous Quantum Field Theory"
Not yet. It is under control in various approximations.
so...imagine for a moment that tomorrow it is mathematically proved that QED in four dimensions can not verify the Wightman Axioms, that is, it is not mathematically consistent. In this hypothetical case, we could not assure that the perturbative calculations and other techniques (as Lattices) would agree (even with super-fast computers). In this hypothetical scenario, maybe the perturbative calculations are the only ones that leads to facts as Lamb shift and anomalous magnetic moment. But anyway, that would not change the fact that "virtual particles" are some mathematical integrals in some computations techniques, so I would not see any fundamental change (relative to the "status" of virtual particles) even in this hypothetical scenario.
Indeed, one can still do QED on the lattice, where virtual particles don't exist.

Note also that the Wightman axioms can at best describe the uncharged sector of QED, thus no real electrons. It is not yet fully understood how the Wightman axioms must be modified to account for the charged particles, though the work by Strocchi and coworkers.

Thus the consistency problem for QED would not be solved by just showing there is no QED satisfying the Wightman axioms. Deciding whether QED is not mathematically consistent is therefore not a very well-defined problem....

#### mattt

Re: "virtual particles" in rigorous quantum field theory

Thank you very much, Arnold.

By the way, your FAQ is great.

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