The potential of the Yukawa theory is unbounded from below for large enough $\phi$. The same is valid for the electrodynamic potential, which is unbounded from below for large enough $A_\mu$. So why these theories are not ruled out?
Because we do perturbation theory around local classical minima. In other words quantum fields are treated as quantum fluctuations around the local classical minima, and Yukawa couplings are taken in the perturbative range h2/4π
The problem involves a combination of the continuous spectrum of fields and the quantum spontaneous production of positive and negative energy particles (while still satisfying conservation laws). There are finite degree-of-freedom systems that are stable despite having interacting positive- and negative-energy degrees of freedom, at least classically. Interactions involve resonant frequencies (see, e.g., Khazin & Shnol, _Stability of Critical Equilibrium States_). For finitely many degrees of freedom, resonances might not be hit, so it seems possible that such theories might be stable even when quantized (though I don't recall seeing it done). But for a field theory (in infinite space), the continuous spectrum implies that there are always wavenumbers such that the resonant frequencies are hit. Thus there will be explosive spontaneous production of positive- and negative-energy particles. At least that is the natural thing to expect. Possible loopholes have been entertained, but would need detailed consideration.
F.F.Faria> So why these theories are not ruled out?
I doubt your swiping statement about unbounded energy.
One relevant system of interest, which has been much studied in recent years, is the Higgs-top quark system, which apparently is near the edge of instability, given the observed parameters. Absent any serious analysis of my own, I take that to imply that the question of stability, in the presence of Yukawa couplings, is a matter of model details. Details which must take into account the fact that all Yukawa couplings are to fermions.
I think it's about stability and also particle creation. I am not sure but I think you are right about Yukawa potential, and it's cause problem of confinement in it's application.
Any way, you can use a general theory with unbounded energy from below but you can't find anything stable and observable. Maybe simplest problem is that system doesn't have vacuum state.
If the energy spectrum is unbounded from below, the system is unstable. So what must be done is define the degrees of freedom that describe a stable system. This doesn't have anything to do with perturbation theory.
1. Unbounded from below potential is not a problem. Unbounded from below spectrum is a problem.
2. Unbounded from below spectrum means instability. Actually it allows something like perpetuum mobile. One could "move" one particle down the spectrum and produce infinite energy.
3. So unbounded from below spectrum produce problems with the second law of thermodynamics. Derivation of the second law of thermodynamics assumes bounded from below spectrum.
4. Actually one can consider unbounded from below spectrum, but in this case one have to guarantee that all effects of this unboundedness are extremely slow. This tricks sometimes are considered in cosmology, some cosmologists consider unbounded from below spectrum of false vacuums.
The clearest answer to the question is answered by MIkhail G. Ivanov. The point is that if the spectrum is unbounded below, the system could collapse into this lowest eigenstate and that would be the end of all physical questions for this system since no finite perturbation of any kind could move thesystem from this lowest energy state.
If the spectrum is unbounded from below there isn't any lowest eigenstate-by definition.
Which brings us back to the original question.
Of course we suspect that somewhere there is a stable state. But it is completely dead and irrelevant. Take Stark effect. We suspect that there electrodes somewhere, we suspect that the electron would eventually leave the atom in the linear potential and the electron and the nucleus would finish up at those electrodes. But that would be quite a different problem. Meanwhile we are interested in the perturbations to the spectrum. Our problem is not correctly stated from the general point of view but the solution is useful.
I think we must be clear about difference between unbounded interaction and unbounded spectrum. In simplest situation, like QM, spectrum calculate from expectation value of potential (besides kinetic term), therefore ground state wave-function is also important. It means that even if potential isn't bounded, it's expectation value can has lower bound with proper wave-function.
I think by Yukawa potential, you mean V(\phi)=a*\phi^4+b*\phi^2, which isn't generally correct, it used as an approximation near spontaneous symmetry breaking situation. It's something like Landau $\phi^4$ theory in phase transition.
The occurrence of infinity in quantum field theory led Dirac in his April 1970 Physics Today article entitled , Can Equation of motion Method be used in High Energy Physics?" based on conventional Lie-algebraic method. This prompted my paper on the implications of non-unitary transformation of Planck's quantum hypothesis. that called for replacement of point-like particles interacting though (divergent) action-at-a-distance forces by extended (non-point-like) particles involving contact interactions (Hulthen potential) and use of generalized (Lie-isotopic and Lie-admissible) equations of motion method. It was successfully applied to the description of the (Rutherford-Santilli) neutron as compressed hydrogen atom.
Azizi> I think by Yukawa potential, you mean V(\phi)=a*\phi^4+b*\phi^2
I have interpreted it to mean a (Yukawa) coupling of the form
g \phi \bar{\psi} \psi.
Which by itself is unbounded in both directions, but one has to consider the complete Lagrangian to decide.
The Yukawa interaction isn't the only term, consistent with the symmetries. The scalar potential is consistent with the symmetries of the theory, therefore there isn't any reason to omit it. For a > 0 (in the notation used) the potential is bounded from below and there isn't any issue-this is known in the literature as the Higgs-Yukawa model. For a
The complete, classical, Lagrangian of fermions interacting with scalars has, along with Yukawa interaction terms, scalar self-potential terms and such Lagrangians have been extensively studied.
If the coefficient of the highest degree in the scalar potential is negative (when the exponent is even) or whatever its sign may be (if the exponent is odd) then the potential is unbounded from below and the theory is unstable. The interaction doesn't make sense and only free fields do.
It can be that, even if the (classical) scalar potential is bounded from below, the theory doesn't make sense, either and can describe only free fields-this is the case above the upper critical dimension, for instance.
The statements about bifurcations don't make sense.
The answer is, first of all, that the expression is incomplete-one needs to include all terms consistent with the symmetries. One such term is λφ4 that is generated by quantum effects, in up to four spacetime dimensions. In three spacetime dimensions one will have a hφ6 term, also. (In two spacetime dimensions all powers are possible, so some more care is needed.) This is the Higgs-Yukawa model, as mentioned above, coupled to a U(1) gauge field, whose kinetic term may need to be included; else there is an external gauge field, i.e. an external electric, or magnetic, field. For strong enough electric fields, the stable vacuum will have a finite density of pairs, for instance. So the instability means precisely that.
It can be studied-and has been studied-by Monte Carlo simulations and by mean field theory and the phase diagram can be found. And there are all sorts of phases. . Only scaling limits define sensible theories and it may be that the only sensible one is the free one, where e and g and all other coupling constants can only take zero values. But it can be that there are special values, different from zero, for certain of the couplings.
All these are words that don't mean anything within the context of the subject. Symmetries can't be defined for a system that doesn't have a ground state at all.
Faria> HYukawa = HDirac + HKlein-Gordon + gψ ̅ψϕ is unbounded from below for large enough −gϕ
Not necessarily, because HKlein-Gordon at least contains a m2 ϕ2 -term. In addition, as Stam said, a ϕ4-term will be generated by quantum effects in the effective theory (and may also be required in the bare theory for a sensible model). The ψ-field is fermionic, and cannot be treated on the same footing as a bosonic field. As I wrote in an earlier post, it is essentially the stability of this model which has been discussed in recent years, in a description of the Higgs-top quark system.
No-the A^4 term is forbidden by gauge invariance. What happens, in an electric field, is that the vacuum of zero density can become unstable, if the field exceeds some threshold, and then the vacuum state of finite density becomes stable. A magnetic field doesn't create pairs, but can have other effects.
One should take care to use gauge invariant quantities.
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