Quantum Field Theory Deserves Extra Help

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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 <P\&Q<\infty $, and obey
$\left[Q\mathrm{,}P\right]\equiv QP-PQ=i\hslash 1\text{l}$. Next we introduce several versions of
$Q\text{\hspace{0.05em}}\left[Q\mathrm{,}P\right]=i\hslash Q$, specifically

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

This equation serves to introduce the “dilation” operator
$D\equiv \left(QP+PQ\right)/2$ ^{1} which leads to
$\left[Q\mathrm{,}D\right]=i\hslash Q$. While
$P\left(={P}^{\u2020}\right)\mathrm{\&}Q\left(={Q}^{\u2020}\right)$ are the foundation of CQ,
$D\left(={D}^{\u2020}\right)\mathrm{\&}Q\left(={Q}^{\u2020}\right)$ are the foundation of AQ. Another way to examine this story is to let
$p\mathrm{,}q\to P\mathrm{,}Q$, while
$d\equiv pq\mathrm{,}q\to D\mathrm{,}Q$.

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
$q\ne \mathrm{0\&}Q\ne 0$. While this may seem to be a problem, it can be very useful to limit such variables, like
$0<q\&Q<\infty $, or
$-\infty <q\&Q<0$, 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

$H\left(\pi \mathrm{,}\phi \right)={\displaystyle \int}\left\{\frac{1}{2}\left[\pi {\left(x\right)}^{2}+{\left(\stackrel{\to}{\nabla}\left(x\right)\right)}^{2}+{m}^{2}\text{\hspace{0.05em}}\phi {\left(x\right)}^{2}\right]+g\phi {\left(x\right)}^{r}\right\}{\text{d}}^{s}x\mathrm{,}$ (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 [2]. Such results are similar for $r=4$ and $n=4$, which is a case where $r=2n/\left(n-2\right)$ [3] [4] [5]. When using AQ, the same models lead to “non-free” results [2] [6].

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 $z\left(x\mathrm{,}t\right)=0$. 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 $\left[\stackrel{^}{\phi}\left(x\right)\mathrm{,}\stackrel{^}{\kappa}\left(y\right)\right]=i\hslash {\delta}^{s}\left(x-y\right)\stackrel{^}{\phi}\left(x\right)$. The affine quantum version of (2) becomes

$\mathcal{H}\left(\stackrel{^}{\kappa}\mathrm{,}\stackrel{^}{\phi}\right)={\displaystyle \int}\left\{\frac{1}{2}\left[\stackrel{^}{\kappa}\left(x\right)\stackrel{^}{\phi}{\left(x\right)}^{-2}\stackrel{^}{\kappa}\left(x\right)+{\left(\stackrel{\to}{\nabla}\stackrel{^}{\phi}\left(x\right)\right)}^{2}+{m}^{2}\stackrel{^}{\phi}{\left(x\right)}^{2}\right]+g\stackrel{^}{\phi}{\left(x\right)}^{r}\right\}{\text{d}}^{s}x\mathrm{.}$ (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<{\left|\stackrel{^}{\phi}\left(x\right)\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 like^{5}

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

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

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

Finally, we let
$a\to 0$ to secure a complete Fourier transformation^{6}

$F\left(f\right)=\mathrm{exp}\left\{-b{\displaystyle \int}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{\text{d}}^{s}x\left(1-{\text{e}}^{if\left(x\right)\phi \left(x\right)}\right){\left|w\left(\phi \left(x\right)\right)\right|}^{2}\text{d}\phi \left(x\right)/\left|\phi \left(x\right)\right|\right\}\mathrm{.}$ (6)

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

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

Observe the factor ${\left|{\phi}_{k}\right|}^{-\left(1-2b{a}^{s}\right)}$ 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

^{1}Even if *Q* does not cover the whole real line, which means that
${P}^{\u2020}\ne P$, yet
${P}^{\u2020}Q=PQ$. This leads to
$D=\left(QP+{P}^{\u2020}Q\right)/2={D}^{\u2020}$.

^{2}For 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 [1].

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

^{4}For 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
$\epsilon ={10}^{-10}$ [2] [6].

^{5}The remainder of this article updates and improves a recent article by the author [7].

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

References

[1] Klauder, J. (2020) Using Affine Quantization to Analyze Non-Renormalizable Scalar Fields and the Quantization of Einsteins Gravity. Journal of High Energy Physics, Gravitation and Cosmology, 6, 802-816.

https://doi.org/10.4236/jhepgc.2020.64053

[2] Fantoni, R. (2021) Monte Carlo Evaluation of the Continuum Limit of . Journal of Statistical Mechanics, 2021, Article ID: 083102.

https://doi.org/10.1088/1742-5468/ac0f69

[3] Freedman, B., Smolensky, P. and Weingarten, D. (1982) Monte Carlo Evaluation of the Continuum Limit of and . Physics Letters B, 113, 481.

https://doi.org/10.1016/0370-2693(82)90790-0

[4] Aizenman, M. (1981) Proof of the Triviality of Field Theory and Some Mean-Field Features of Ising Models for . Physical Review Letters, 47, 1-4.

https://doi.org/10.1103/PhysRevLett.47.1

[5] Fröhlich, J. (1982) On the Triviality of Theories and the Approach to the Critical Point in Dimensions. Nuclear Physics B, 200, 281-296.

https://doi.org/10.1016/0550-3213(82)90088-8

[6] Fantoni, R. and Klauder, J. (2021) Affine Quantization of Succeeds While Canonical Quantization Fails. Physical Review D, 103, Article ID: 076013.

https://doi.org/10.1103/PhysRevD.103.076013

[7] Klauder, J. (2021) Evidence for Expanding Quantum Field Theory. Journal of High Energy Physics, Gravitation and Cosmology, 7, 1157-1160.

https://doi.org/10.4236/jhepgc.2021.73067