Quantum Field Theory Deserves Extra Help

Abstract

Today's quantum field theory (QFT) relies heavenly on canonical quantization (CQ), which fails for φ44 leading only to a “free” result. Affine quantization (AQ), an alternative quantization procedure, leads to a “non-free” result for the same model. Perhaps adding AQ to CQ can improve the quantization of a wide class of problems in QFT.

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Klauder, J. (2022) Quantum Field Theory Deserves Extra Help. Journal of High Energy Physics, Gravitation and Cosmology, 8, 265-268. doi: 10.4236/jhepgc.2022.82021.

1. What is AQ?

The simplest way to understand AQ is to derive it from CQ. The classical variables, p & q, lead to self-adjoint quantum operators, P & Q, that cover the real line, i.e., $-\infty , and obey $\left[Q,P\right]\equiv QP-PQ=i\hslash 1\text{l}$. Next we introduce several versions of $Q\text{ }\left[Q,P\right]=i\hslash Q$, specifically

$\begin{array}{l}\left\{Q\left[Q,P\right]+\left[Q,P\right]Q\right\}/2=\left\{{Q}^{2}P-QPQ+QPQ-P{Q}^{2}\right\}/2\\ =\left\{Q\left(QP+PQ\right)-\left(QP+PQ\right)Q\right\}/2=\left[Q,QP+PQ\right]/2.\end{array}$ (1)

This equation serves to introduce the “dilation” operator $D\equiv \left(QP+PQ\right)/2$ 1 which leads to $\left[Q,D\right]=i\hslash Q$. While $P\left(={P}^{†}\right)&Q\left(={Q}^{†}\right)$ are the foundation of CQ, $D\left(={D}^{†}\right)&Q\left(={Q}^{†}\right)$ are the foundation of AQ. Another way to examine this story is to let $p,q\to P,Q$, while $d\equiv pq,q\to D,Q$.

Observe, for CQ, that while q & Q range over the whole real line, that is not possible for AQ. If $q\ne 0$ then d covers the real line, but if $q=0$ then $d=0$ and p is helpless. To eliminate this possibility we require $q\ne 0&Q\ne 0$. While this may seem to be a problem, it can be very useful to limit such variables, like $0, or $-\infty , or even both.2

2. A Look at Quantum Field Theory

2.1. Selected Poor and Good Results

Classical field theory normally deals with a field $\phi \left(x\right)$ and a momentum $\pi \left(x\right)$, where x denotes a spatial point in an underlying space.3

A common model for the Hamiltonian is given by

$H\left(\pi ,\phi \right)=\int \left\{\frac{1}{2}\left[\pi {\left(x\right)}^{2}+{\left(\stackrel{\to }{\nabla }\left(x\right)\right)}^{2}+{m}^{2}\text{ }\phi {\left(x\right)}^{2}\right]+g\phi {\left(x\right)}^{r}\right\}{\text{d}}^{s}x,$ (2)

where $r\ge 2$ is the power of the interaction term, $s\ge 2$ is the dimension of the spatial field, and $n=s+1$, which adds the time dimension. Using CQ, such a model is nonrenormalizable when $r>2n/\left(n-2\right)$, which leads to “free” model results . Such results are similar for $r=4$ and $n=4$, which is a case where $r=2n/\left(n-2\right)$   . When using AQ, the same models lead to “non-free” results  .

Solubility of classical models involves only a single path, while quantization involves a vast number of paths, a fact well illustrated by path-integral quantization. The set of acceptable paths can shrink significantly when a nonrenormalizable term is introduced. Divergent paths of integration are like those for which $\phi \left(x,t\right)=1/z\left(x,t\right)$ when $z\left(x,t\right)=0$. A procedure that forbids possibly divergent paths would eliminate nonrenormalizable behavior. As we note below, AQ provides such a procedure.

2.2. The Classical and Quantum Affine Story

Classical affine field variables are $\kappa \left(x\right)\equiv \pi \left(x\right)\phi \left(x\right)$ and $\phi \left(x\right)\ne 0$. The quantum versions are $\stackrel{^}{\kappa }\left(x\right)\equiv \left[\stackrel{^}{\phi }\left(x\right)\stackrel{^}{\pi }\left(x\right)+\stackrel{^}{\pi }\left(x\right)\stackrel{^}{\phi }\left(x\right)\right]/2$ and $\stackrel{^}{\phi }\left(x\right)\ne 0$, with $\left[\stackrel{^}{\phi }\left(x\right),\stackrel{^}{\kappa }\left(y\right)\right]=i\hslash {\delta }^{s}\left(x-y\right)\stackrel{^}{\phi }\left(x\right)$. The affine quantum version of (2) becomes

$\mathcal{H}\left(\stackrel{^}{\kappa },\stackrel{^}{\phi }\right)=\int \left\{\frac{1}{2}\left[\stackrel{^}{\kappa }\left(x\right)\stackrel{^}{\phi }{\left(x\right)}^{-2}\stackrel{^}{\kappa }\left(x\right)+{\left(\stackrel{\to }{\nabla }\stackrel{^}{\phi }\left(x\right)\right)}^{2}+{m}^{2}\stackrel{^}{\phi }{\left(x\right)}^{2}\right]+g\stackrel{^}{\phi }{\left(x\right)}^{r}\right\}{\text{d}}^{s}x.$ (3)

The spacial differential term restricts $\stackrel{^}{\phi }\left(x\right)$ to continuous operator functions, maintaining $\stackrel{^}{\phi }\left(x\right)\ne 0$. In that case, it follows that $0<\stackrel{^}{\phi }{\left(x\right)}^{-2}<\infty$ which implies that $0<{|\stackrel{^}{\phi }\left(x\right)|}^{r}<\infty$ for all $r<\infty$, a most remarkable feature because it forbids nonrenormalizability!4

Adopting a Schrödinger representation, where $\stackrel{^}{\phi }\left(x\right)\to \phi \left(x\right)$, simplifies $\stackrel{^}{\kappa }\left(x\right)\phi {\left(x\right)}^{-1/2}=0$, which also implies that $\stackrel{^}{\kappa }\left(x\right){\Pi }_{y}\phi {\left(y\right)}^{-1/2}=0$. This relation suggests that a general wave function is like $\Psi \left(\phi \right)=W\left(\phi \right){\Pi }_{y}\phi {\left(y\right)}^{-1/2}$, as if ${\Pi }_{y}\phi {\left(y\right)}^{-1/2}$ acts as the representation of a family of similar wave functions.

We now take a Fourier transformation of the absolute square of a regularized wave function that looks like5

$F\left(f\right)={\Pi }_{k}\int \left\{{\text{e}}^{i{f}_{k}{\phi }_{k}}{|w\left({\phi }_{k}\right)|}^{2}\left(b{a}^{s}\right){|{\phi }_{k}|}^{-\left(1-2b{a}^{s}\right)}\text{d}{\phi }_{k}\right\}.$ (4)

Normalization ensures that if all ${f}_{k}=0$, then $F\left(0\right)=1$, which leads to

$F\left(f\right)={\Pi }_{k}\int \left\{1-\int \left(1-{\text{e}}^{i{f}_{k}{\phi }_{k}}\right){|w\left({\phi }_{k}\right)|}^{2}\left(b{a}^{s}\right)\text{d}{\phi }_{k}/{|{\phi }_{k}|}^{\left(1-2b{a}^{s}\right)}\right\}.$ (5)

Finally, we let $a\to 0$ to secure a complete Fourier transformation6

$F\left(f\right)=\mathrm{exp}\left\{-b\int \text{ }\text{ }{\text{d}}^{s}x\left(1-{\text{e}}^{if\left(x\right)\phi \left(x\right)}\right){|w\left(\phi \left(x\right)\right)|}^{2}\text{d}\phi \left(x\right)/|\phi \left(x\right)|\right\}.$ (6)

This particular process side-steps any divergences that may normally arise in $|w\left(\phi \left(x\right)\right)|$ when using more traditional procedures.

3. The Absence of Nonrenormalizablity, and the Next Fourier Transformation

Observe the factor ${|{\phi }_{k}|}^{-\left(1-2b{a}^{s}\right)}$ in (4) which is prepared to insert a zero divergence for each and every ${\phi }_{k}$ when $a\to 0$. However, the factor $b{a}^{s}$ in (4) turns that possibility into a very different story given in (6).

Another Fourier transformation can take us back to a suitable function of the field, $\phi \left(x\right)$. That task involves pure mathematics, and it deserves a separate analysis of its own.

NOTES

1Even if Q does not cover the whole real line, which means that ${P}^{†}\ne P$, yet ${P}^{†}Q=PQ$. This leads to $D=\left(QP+{P}^{†}Q\right)/2={D}^{†}$.

2For example, affine quantization of gravity can restrict operator metrics to positivity,i.e., ${\stackrel{^}{g}}_{ab}\left(x\right)d{x}^{a}d{x}^{b}>0$, straight away .

3In order to avoid problems with spacial infinity we restrict our space to the surface of a large, $\left(s+1\right)$ -dimensional sphere.

4For Monte Carlo studies, concern for the term $\stackrel{^}{\phi }{\left(x\right)}^{-2}\ne 0$ has been resolved by successful usage of ${\left[\stackrel{^}{\phi }{\left(x\right)}^{2}+\epsilon \right]}^{-1}$, where $\epsilon ={10}^{-10}$  .

6Any change of $w\left(\phi \right)$ due to $a\to 0$ is left implicit.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

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