Today's quantum field theory (QFT) relies heavenly on canonical quantization (CQ), which fails for φ 4 4 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.
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 ℏ 1 l . Next we introduce several versions of Q [ Q , P ] = i ℏ 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 ℏ 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
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 [
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.
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 ℏ 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.
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.
The author declares no conflicts of interest regarding the publication of this paper.
Klauder, J.R. (2022) Quantum Field Theory Deserves Extra Help. Journal of High Energy Physics, Gravitation and Cosmology, 8, 265-268. https://doi.org/10.4236/jhepgc.2022.82021