Canonical quantization (CQ) is built around [ Q, P] = iħ1l , while affine quantization (AQ) is built around [ Q, D] = iħQ, where D ≡ ( PQ + QP) / 2 . The basic CQ operators must fit -∞ < P, Q < ∞ , while the basic AQ operators can fit -∞ < P < ∞ and 0 < Q < ∞ , -∞ < Q < 0 , or even -∞ < Q ≠ 0 < ∞ . AQ can also be the key to quantum gravity, as our simple outline demonstrates.
The CQ equation [ Q , P ] = Q P − P Q = i ℏ 1 l is multiplied by Q to give { Q ( Q P − P Q ) + ( Q P − P Q ) Q } / 2 = { Q 2 P + Q P Q − Q P Q − P Q 2 } / 2 = [ Q , ( Q P + P Q ) / 2 ] ≡ [ Q , D ] = i ℏ Q .
This principal equation requires that Q ≠ 0 , leading to the last expression which applies for Q > 0 , Q < 0 , or both. From a classical promotion viewpoint for CQ it is − ∞ < p → P , q → Q < ∞ , while for AQ it is − ∞ < d ≡ p q → D ≡ ( P Q + Q P ) / 2 < ∞ , while − ∞ < q ≠ 0 → Q ≠ 0 < ∞ . Note: P † ≠ P , but P † Q = P Q .
Not every set of classical variables lead to valid quantum operators, neither for CQ nor for AQ. The correct choice of classical variables to promote quantum operators for CQ is Cartesian coordinates, which create a constant vanishing curvature. The classical coordinates that create a constant negative curvature1 are the correct coordinates to promote quantum operators for AQ.2 Regarding CQ and AQ operators, the principal difference is that CQ operators must span the whole real line, while the AQ operator Q strictly spans all positive coordinates, all negative coordinates, or both, provided Q ≠ 0 .3
Different sets of classical problems require valid CQ or AQ quantizations. For example, a traditional harmonic oscillator, with − ∞ < p , q < ∞ , employs CQ, while a half-harmonic oscillator, with − ∞ < p < ∞ , but 0 < q < ∞ , employs AQ [
Examples from scalar field theories show that φ 3 12 leads to “free” results for CQ while “non-free” results for AQ [
Regarding gravity, an affine result can offer a strictly positive metric, i.e., d s ( x ) 2 = g a b ( x ) d x a d x b > 0 , provided d x c ≡ 0 . This ability offers a strong contribution to the quantization of gravity, as sketched below.
The author has shown that an affine quantization is the natural procedure to quantize Einstein’s gravity, e.g., [
The classical Hamiltonian, according to ADM [
H ( π b a , g c d ) = ∫ { g ( x ) − 1 / 2 [ π b a ( x ) π a b ( x ) − 1 2 π a a ( x ) π b b ( x ) ] + g ( x ) 1 / 2 R ( x ) } d 3 x , (1)
where R ( x ) is the Ricci scalar [
{ π b a ( x ) , π d c ( x ′ ) } = 1 2 δ 3 ( x , x ′ ) [ δ d a π b c ( x ) − δ b c π d a ( x ) ] , { g a b ( x ) , π d c ( x ′ ) } = 1 2 δ 3 ( x , x ′ ) [ δ a c g b d ( x ) + δ b c g a d ( x ) ] , { g a b ( x ) , g c d ( x ′ ) } = 0. (2)
Observe that these Poisson brackets are valid even if we change g a b ( x ) to − g a b ( x ) , a property that allows us to retain only g a b ( x ) > 0 . This is not possible with the Poisson brackets for canonical variables.
Passing to operator commutations by a promotion of the set of Poisson brackets to operator commutations leads, as established in [
[ π ^ b a ( x ) , π ^ d c ( x ′ ) ] = i 1 2 ℏ δ 3 ( x , x ′ ) [ δ d a π ^ b c ( x ) − δ b c π ^ d a ( x ) ] , [ g ^ a b ( x ) , π ^ d c ( x ′ ) ] = i 1 2 ℏ δ 3 ( x , x ′ ) [ δ a c g ^ b d ( x ) + δ b c g ^ a d ( x ) ] , [ g ^ a b ( x ) , g ^ c d ( x ′ ) ] = 0. (3)
As with the Poisson brackets, these commutators are valid if we change g ^ a b ( x ) to − g ^ a b ( x ) . For the momentric and metric operators, we again find that we can retain only g ^ a b ( x ) > 0 . Moreover, with suitable regularization, it follows that π ^ b a ( x ) g ^ ( x ) − 1 / 2 = 0 , which is proved in [
Schrödinger’s representation chooses g ^ a b ( x ) = g a b ( x ) and
π ^ d c ( x ) = − i 1 2 ℏ [ g d e ( x ) ( ∂ / ∂ g c e ( x ) ) + ( ∂ / ∂ g c e ( x ) ) g d e ( x ) ] , (4)
which is well evaluated using a suitable regularization [
i ℏ ∂ Ψ ( { g } , t ) / ∂ t = { ∫ { [ π ^ b a ( x ) g ( x ) − 1 / 2 π ^ a b ( x ) − 1 2 π ^ a a ( x ) g ( x ) − 1 / 2 π ^ b b ( x ) ] + g ( x ) 1 / 2 R ( x ) } d 3 x } Ψ ( { g } , t ) . (5)
The expression π ^ b a ( x ) g ( x ) − 1 / 2 = 0 points to a form of wave function given by Ψ ( { g } ) = W ( { g } ) Π y g ( y ) − 1 / 2 . The wave functions for the Hamiltonian are continuous in the metrics that make up { g } , which is natural for a Hamiltonian that contains spatial derivatives of the metric fields and enforces continuity within the Ricci scalar, R ( x ) .
The normalization of such wave functions may be given by
∫ | Ψ ( { g } ) | 2 D { g } = ∫ | W ( { g } ) | 2 Π y g ( y ) − 1 D { g } = 1, (6)
while a matrix expression for an operator A may be given by
∫ Ψ ′ ( { g ′ } ) * A ( { g ′ } , { g } ) Ψ ( { g } ) D { g ′ } D { g } = ∫ Π y ′ g ′ ( y ′ ) − 1 / 2 A ( { g ′ } , { g } ) Π y g ( y ) − 1 / 2 D { g ′ } D { g } . (7)
It may be noticed that the factors Π y g ( y ) − 1 / 2 act somewhat like coherent states’, e.g., [
The forgoing study is focused on the gravity Hamiltonian, which is the most difficult aspect of quantum gravity. Additional topics, such as fulfilling constraints, are examined in [
Contributions by other researchers using affine procedures to quantize gravity are welcome to add to the present story.
The author declares no conflicts of interest regarding the publication of this paper.
Klauder, J.R. (2021) A Simple Factor in Canonical Quantization Yields Affine Quantization Even for Quantum Gravity. Journal of High Energy Physics, Gravitation and Cosmology, 7, 1328-1332. https://doi.org/10.4236/jhepgc.2021.74082