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 JHEPGC  Vol.8 No.2 , April 2022
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.

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., < P & Q < , and obey [ Q , P ] Q P P Q = i 1 l . Next we introduce several versions of Q [ Q , P ] = i Q , specifically

{ Q [ Q , P ] + [ Q , P ] Q } / 2 = { Q 2 P Q P Q + Q P Q P Q 2 } / 2 = { Q ( Q P + P Q ) ( Q P + P Q ) Q } / 2 = [ Q , Q P + P Q ] / 2 . (1)

This equation serves to introduce the “dilation” operator D ( Q P + P Q ) / 2 1 which leads to [ Q , D ] = i Q . While P ( = P ) & Q ( = Q ) are the foundation of CQ, D ( = D ) & Q ( = Q ) are the foundation of AQ. Another way to examine this story is to let p , q P , Q , while d p q , q D , Q .

Observe, for CQ, that while q & Q range over the whole real line, that is not possible for AQ. If q 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 0 & Q 0 . While this may seem to be a problem, it can be very useful to limit such variables, like 0 < q & Q < , or < 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 φ ( x ) and a momentum π ( x ) , where x denotes a spatial point in an underlying space.3

A common model for the Hamiltonian is given by

H ( π , φ ) = { 1 2 [ π ( x ) 2 + ( ( x ) ) 2 + m 2 φ ( x ) 2 ] + g φ ( x ) r } d s x , (2)

where r 2 is the power of the interaction term, s 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 > 2 n / ( n 2 ) , which leads to “free” model results [2]. Such results are similar for r = 4 and n = 4 , which is a case where r = 2 n / ( n 2 ) [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 φ ( x , t ) = 1 / z ( x , t ) when z ( x , t ) = 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 κ ( x ) π ( x ) φ ( x ) and φ ( x ) 0 . The quantum versions are κ ^ ( x ) [ φ ^ ( x ) π ^ ( x ) + π ^ ( x ) φ ^ ( x ) ] / 2 and φ ^ ( x ) 0 , with [ φ ^ ( x ) , κ ^ ( y ) ] = i δ s ( x y ) φ ^ ( x ) . The affine quantum version of (2) becomes

H ( κ ^ , φ ^ ) = { 1 2 [ κ ^ ( x ) φ ^ ( x ) 2 κ ^ ( x ) + ( φ ^ ( x ) ) 2 + m 2 φ ^ ( x ) 2 ] + g φ ^ ( x ) r } d s x . (3)

The spacial differential term restricts φ ^ ( x ) to continuous operator functions, maintaining φ ^ ( x ) 0 . In that case, it follows that 0 < φ ^ ( x ) 2 < which implies that 0 < | φ ^ ( x ) | r < for all r < , a most remarkable feature because it forbids nonrenormalizability!4

Adopting a Schrödinger representation, where φ ^ ( x ) φ ( x ) , simplifies κ ^ ( x ) φ ( x ) 1 / 2 = 0 , which also implies that κ ^ ( x ) Π y φ ( y ) 1 / 2 = 0 . This relation suggests that a general wave function is like Ψ ( φ ) = W ( φ ) Π y φ ( y ) 1 / 2 , as if Π y φ ( y ) 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 ( f ) = Π k { e i f k φ k | w ( φ k ) | 2 ( b a s ) | φ k | ( 1 2 b a s ) d φ k } . (4)

Normalization ensures that if all f k = 0 , then F ( 0 ) = 1 , which leads to

F ( f ) = Π k { 1 ( 1 e i f k φ k ) | w ( φ k ) | 2 ( b a s ) d φ k / | φ k | ( 1 2 b a s ) } . (5)

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

F ( f ) = exp { b d s x ( 1 e i f ( x ) φ ( x ) ) | w ( φ ( x ) ) | 2 d φ ( x ) / | φ ( x ) | } . (6)

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

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

Observe the factor | φ k | ( 1 2 b a s ) in (4) which is prepared to insert a zero divergence for each and every φ k when a 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, φ ( x ) . 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 P , yet P Q = P Q . This leads to D = ( Q P + P Q ) / 2 = D .

2For example, affine quantization of gravity can restrict operator metrics to positivity,i.e., g ^ a b ( x ) d x a d x b > 0 , straight away [1].

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

4For Monte Carlo studies, concern for the term φ ^ ( x ) 2 0 has been resolved by successful usage of [ φ ^ ( x ) 2 + ε ] 1 , where ε = 10 10 [2] [6].

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

6Any change of w ( φ ) due to a 0 is left implicit.

Cite this paper: 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.
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

 
 
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