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Dirac’s rule in which only special phase space variables should be promoted to operators in canonical quantization is applied to loop quantum gravity. For this theory, Dirac’s rule is violated, and as a result loop quantum gravity fails the test to be a valid quantization. Indications are included on how to create and deal with valid versions of quantum gravity.

For a single degree of freedom, a momentum p and a position q, where − ∞ < p , q < ∞ , the Poisson bracket is { q , p } = 1 , and the Hamiltonian function is given by H ( p , q ) . In addition, new variables^{1} may also be used, say, p ¯ and q ¯ , { q ¯ , p ¯ } = 1 , − ∞ < p ¯ , q ¯ < ∞ , and H ¯ ( p ¯ , q ¯ ) = H ( p , q ) .

For canonical quantization, we promote p → P , q → Q , [ Q , P ] = i ℏ , along with H ( p , q ) → H ( P , Q ) . In addition, p ¯ → P ¯ , q ¯ → Q ¯ , [ Q ¯ , P ¯ ] = i ℏ , and H ¯ ( p ¯ , q ¯ ) → H ¯ ( P ¯ , Q ¯ ) , BUT, H ¯ ( P ¯ , Q ¯ ) ≠ H ( P , Q ) . At most, only one such quan- tization can be valid while all others lead to false quantizations.

Although the classical Hamiltonians can be equal the quantum Hamiltonians are different, and the question arises which is the physically correct Hamiltonian operator. Dirac [^{2}.

These variables enjoy d p ∧ d q as measures of the appropriate phase space. The same can be said about ∫ { d π ( x ) ∧ d ϕ ( x ) } d x .

Using canonical quantization, the case of loop quantum gravity involves two sets of fields classically denoted by E i a ( x ) and A a i ( x ) , where a , i = 1 , 2 , 3 , and x denotes a 3-dimensional spatial point in space. These variables admit the phase- space measure ∫ { d A a i ( x ) ∧ d E i a ( x ) } d x . However, their natural metric expressions, such as d σ ( A , E ) 2 = ∫ [ C ( x ) ( E i a ( x ) d A a i ( x ) ) 2 + C ( x ) − 1 ( A a i ( x ) d E i a ( x ) ) 2 ] d x , where 0 < C ( x ) < ∞ , fail to exhibit suitable Cartesian coordinates, and thus signal a false quantization because it does not follow Dirac’s rule.

What is affine quantization? While canonical quantization employs Q and P, with [ Q , P ] = i ℏ , as basic operators, affine quantization employs Q and D ≡ 1 2 ( P Q + Q P ) , the dilation operator, with [ Q , D ] = i ℏ Q ; note: the operator D can be self-adjoint even when Q > 0 is self-adjoint, but then P can not be self-adjoint.

There are some systems that canonical quantization can solve, and there are some systems that affine quantization can solve. If they solve using one system they typically fail to solve using the other system. For example, canonical quantization

can solve the Hamiltonian H = 1 2 ( p 2 + q 2 ) , where − ∞ < p , q < ∞ , while affine can not solve it. On the other hand, the same Hamiltonian, H = 1 2 ( p 2 + q 2 ) , now with − ∞ < p < ∞ and 0 < q < ∞ , can be solved with affine quantization but not with canonical quantization. This example is used to illustrate the power of affine quantization in [

Articles [

The representations of the analysis in these two papers may be different, but the physics is the same: specifically, for example, the quantum gravitational metrics are not discrete, but continuous.

The authors declare no conflicts of interest regarding the publication of this paper.

Klauder, J.R. (2020) Is Loop Quantum Gravity a Physically Correct Quantization? Journal of High Energy Physics, Gravitation and Cosmology, 6, 49-51. https://doi.org/10.4236/jhepgc.2020.61006