# An alternative, algebraic way to introduce interactions. Are there other ways out there?

An opening paragraph:

The usual approach to introducing interactions in quantum field theory is to make the constraint on the amplitude of the field towards smaller values more forceful than harmonic. For example, instead of the massive Klein-Gordon field's harmonic constraint towards smaller field values, $\partial_\mu\partial^\mu\phi(x)+m^2\phi(x)=0$, we introduce a more forceful constraint, $\partial_\mu\partial^\mu\phi(x)+\lambda_1\phi(x)+\lambda_3\phi(x)^3=0$. We require $\lambda_3>0$; $\lambda_1$ may be tuned so that a non-zero value of $\phi(x)$ is preferred, if $\lambda_1<0$. Here, we instead introduce a constraint on the observed field by an alternative, algebraic method. We take the observed field to be a bounded function of the field, $\Phi(x)=\mathsf{gd}(\phi(x))$, where the Gudermannian, $$\mathsf{gd}(z)=2\tan{\!{}^{-1}\!\left(e^z\right)}-\frac{\pi}{2}=\tan{\!{}^{-1}\!\left(\sinh{z}\right)},$$ amongst other possible definitions, is complex analytic for $z\not=(2n+1)\pi i$ and has bounded real component between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$. The use of the Gudermannian is inessential to the development, but we will use it as a convenient explicit mapping of the infinite range of measurement values of a free (quantum) field $\hat\phi(x)$ into a bounded range of measurement values $\mathsf{gd}(\hat\phi(x))$, with which we can make some analytic progress. In the usual terms of a potential energy, this construction effectively imposes an infinite potential on the field outside the range $(-\frac{\pi}{2},\frac{\pi}{2})$, because field values outside this range cannot occur, while preserving an attractive harmonic potential when the field is close to zero. It seems possible that the process of renormalization in the context of Hamiltonian or Lagrangian formalisms, by infinite adjustments of parameters, effectively constructs a potential that is comparable to this analytic ideal.

My doubts about this paragraph are a little worrying. It emerged from the mathematics that I've been pursuing for the last two months, building on other mathematics that goes back about a year. I'm quite embedded in a way of thinking that is somewhat different from most peoples' conventional. When in this state, something like the above always looks convincing to me —I discover six months later why it's not, but right now I'm rather (too) pleased with what you see above. I'm asking here (1) whether people see the paragraph above as plausible enough at least to withhold judgment until they see the Math, or even that it seems enticing; or (2) that it looks hopeless (hopefully including some hint as to why, of course); and (3) whether there's anything that this reminds you of [Citations welcome, in the Math or Physics literature]. Answers that let me shortcut around that six month wait to seeing why this can't work will be welcome.

I guess this is, as the sidebar recommends, "sharing my research", though I don't want to take the space to "provide details" beyond a certain point. I'm curious whether it's an effective question. In formulating this question around the above paragraph, it occurs to me to clarify that I'm unhappy with deformation of a Hamiltonian or Lagrangian formulation of a quantum field as the only way we have to construct a new quantum field, particularly because of renormalization, and I'm also unhappy with the nonconstructive tendency in axiomatic quantum field theory. I don't see anything else in the literature, but is there?

EDIT: Luboš rightly focuses on the unlikeliness of the Gudermannian as a preferred choice of transform. As little as possible should depend on the choice of field coordinatization. The working abstract I have calls it a toy model, which I would hope would bring the comment above into focus, that

"The use of the Gudermannian is inessential to the development".

Nonetheless, the Gudermannian is mathematically useful in the QFT context I am developing (1) because it has poles only on the imaginary axis, so just two series expansions are enough to evaluate it, effectively around $\pm\infty$, which is of course possible only because the Gudermannian is finite at $\pm\infty$. This makes analysis considerably more manageable. [Explicitly, we can express $\mathsf{gd}(z)=\mp 2\sum\limits_{j=0}^\infty(-1)^j\frac{1-e^{\pm(2j+1)z}}{2j+1}$ for $\Re(z)<0$ and $\Re(z)>0$.]

(2) Using $\mathsf{gd}$ instead of $\arctan{}$ is also useful for analysis, because the exponential of the quantum field acts as a displacement operator instead of as a derivation. [Explicitly, $[a(x),f(a^\dagger(y))]=f\mathbf{\,'}(a^\dagger)C(x-y)$, where $[a(x),a^\dagger(y)]=C(x-y)$ is the commutator, acts as a derivation, whereas $e^{\alpha a(x)}f(a^\dagger(y))=f(a^\dagger(y)+\alpha C(x-y))e^{\alpha a(x)}$ acts as a displacement operator.]

(3) $\tanh{}$ is a superficially simpler choice, but it's significantly less convergent. [Explicitly, we have to expand the denominator of $\frac{-1+e^{2z}}{1+e^{2z}}$.] $\mathsf{gd}$ is the simplest bounded function that satisfies these three criteria. I so far can't think of a way to get all this into an opening paragraph, unless there is a precedent I don't know of that sets up a context that is close enough to what I'm doing.

I find even the little I can do with this toy model gives a little insight into renormalization, though it's certainly allusive. There is a chaotic regime at time-like separation, for example. Perhaps I have to get into the paragraph above that the move to analysis might be worth trying just because perturbation theory is a non-convergent expansion. I'm already seeing things slightly different because of your comments. Many thanks.

EDIT(2): Luboš asks for clarification of the general idea. At the moment I'm trying to see ways to combine multiple heterodox ways of constructing models. To me, the Hilbert space structure is paramount because it underlies any probability interpretation; positivity of the energy is less important than it is usually taken to be, because it is neither necessary or sufficient to ensure stability; a continuous mass spectrum arises in the Källén-Lehmann representation of the 2-point function of a renormalized theory, so it should not be ruled out as part of other types of models; field renormalization suggests some kind of transformation of the unbounded quantum field to bounded values. Latterly, I've thought that a more analytic approach to models would be better than the series expansions that are universally used. In a few weeks or years I may have dropped any or all of these constituents of the models I have been constructing and characterizing. The published papers that are referred to on my web-site gives some idea of the different kinds of heterodox models I've constructed, understood at some level, and dismissed in the past.

There is an answer, by the way, to the question "Are there other ways out there?", which I came across only a few days ago. Detlev Buchholz, Gandalf Lechner, Stephen J. Summers, "Warped Convolutions, Rieffel Deformations and the Construction of Quantum Field Theories". This and its references claim to construct an interacting QFT in 3+1 in the Haag-Kastler axiomatic context, although I'm just now making my way through it.

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I don't understand why this is an alternative way to introduce interactions. You're just guessing a potential for a scalar field in a theory - your theory because this random function is almost certainly not a potential in any theory we care about (why it should be?). And the only special feature of this function is that it is bounded. Well, it's pretty normal for realistic potentials e.g. in string theory to converge to a constant when the potential is sent to $\pm\infty$. The dilaton is a typical example. –  Luboš Motl Mar 3 '11 at 6:45
However, in a theory with gravity, it's questionable what a "constant potential" is because one may always change the units - the "frame" - by redefining the metric by a multiplicative function of the dilaton (or other similar scalar fields). But in a general (effective) field theory, it's surely the case that one may add rather general potentials of scalar fields to the Lagrangian, although they're unlikely to be a randomly guessed function, and these potentials may also be viewed as results of resummations including loops. –  Luboš Motl Mar 3 '11 at 6:47
I will probably think of why this comment is obviously inappropriate later, but a first thought is about that Gudermannian transform (which I hadnt heard of before). It looks a little bit like a Conformal Transformation (of the $\phi$ field). If so then there is potential here (in several senses). –  Roy Simpson Mar 3 '11 at 11:19
@Roy @Luboš is right that picking off the Gudermannian as special is not a good idea as Physics. The working abstract I have calls it a Toy Model. I've added a clarification in the Question, although what should go into an effective first paragraph currently eludes me. –  Peter Morgan Mar 3 '11 at 14:19
You are not introducing interactions, you are reparametrizing a field. –  Ron Maimon Sep 4 '11 at 4:49