Quantum Gravity Made Easy

Gravity does not naturally fit well with canonical quantization. Affine quantization is an alternative procedure that is similar to canonical quantization but may offer a positive result when canonical quantization fails to offer a positive result. Two simple examples given initially illustrate the power of affine quantization. These examples clearly point toward an affine quantization procedure that vastly simplifies a successful quantization of general relativity.


Introduction
In order to offer a credible analysis of quantum gravity, it is first necessary to carefully review several common questions: 1) Are the rules of canonical quantization the full story of how to quantize any particular classical theory? 2) Is the standard assumption that the correct set of basic, phase space classical variables to promote to operator variables are Cartesian coordinates? 3) How do we choose Cartesian, phase space coordinates when phase space has no metric? 4) Is it necessary when taking the classical limit of a quantum theory to choose As usual, Q and P are irreducible, but Q and D are reducible in that D and 0 Q > is irreducible along with D and 0 Q < ; a third case where 0 Q = is less important. If 0 Q > (or 0 Q < ), then P cannot be made self adjoint; however, in that case, both Q and D are self adjoint. The operator D is called the dilation operator because it dilates Q rather than translates Q as P does; in particular, ( where in the second relation, 0 q ≠ , and q, as well as Q, are normally chosen to be dimensionless. Observe: According to (1), the existence of canonical operators guarantees the existence of affine operators!

Canonical and Affine Coherent States
The canonical coherent states are well known and generally given by where the fiducial vector 0 satisfies ( These vectors admit a resolution of unity given by The affine coherent states are less well known and they are generally given, for where the fiducial vector β satisfies ( ) ( ) ; we choose a common notation for the two sets of coherent states, but the different range of variables helps set them apart. The affine vectors admit a resolution of unity given by −  , which requires that 2 β >  . A similar story applies to 0 q < and 0 Q < , or a combination so that 0 q ≠ , but we focus on 0 q > which has more relevance for gravity.

Classical/Quantum Connection
and the variational result is given by For the affine story, we use affine coherent states which lead to Equation (12) applies as well when 0 q ≠ , which makes it more similar to Equation (11).
Notice that the canonical and affine versions of the reduced action functionals are effectively identical in that they both appear as classical action functionals! In fact, they are "better" than classical expressions because they still involve  which is not zero. To recover the usual classical story from the quantum story in conventional canonical quantization requires that 0 →  , but that is highly unphysical because the world we all live in is one where 0 >  . Indeed, we prefer to refer to the Hamiltonians in (11) and (12)) as enhanced classical Hamiltonians because they each retain 0 >  .
The expressions for the enhanced classical actions for both the canonical and affine stories have the property that if phase space coordinates are changed, such because the original point in phase space must be mapped to the same vector in Hilbert space. This property ensures that even though the phase space variables are changed all of the quantum aspects remain unchanged; this favorable property works whether the underlying operators are canonical or affine.
The common behavior of the enhanced classical stories implies that a classical theory can be quantized by either canonical or affine procedures with the same justification. If one approach fails, try the other one!

''Cartesian Coordinates"
Canonical quantization "promotes" classical variables to operators, e.g., p P → and q Q → , and builds its operator Hamiltonian from But which pair of classical variables should be promoted to operators. The standard answer to this question is that the proper classical phase space variables should be "Cartesian coordinates", according to Dirac [3] (page 114, in a footnote).
Enhanced quantization offers a clear connection between quantum and classical variables. For the canonical case, the enhanced classical Hamiltonian is given by and offers a flat space that already involves Cartesian coordinates, thereby confirming Dirac's rule! For the case of affine variables, with 0 q > as our example, the enhanced classical Hamiltonian is given by Thus, with a polynomial Hamiltonian for clarity, and in the normal classical and is geodesically complete [5]! Observed that the "quadratic coefficient" terms in (17) yield unity when they are multiplied together; this property for affine metrics will rise again. The

An Example
The harmonic oscillator with the classical Hamiltonian where ( ) 2 , p q ∈  , and its canonical quantization is so well known we rely on the reader for its behavior; moreover, for this example, we concede that canonical quantization beats affine quantization. However, the identical classical Hamiltonian which is now restricted so that 0 q > , i.e., ( ) as expected.

Schrödinger's Representation and Equation
The Schrödinger representation for 0 Q > is 0 x > , where x is simply a positive real number, and for D is ( ) ( ) ( ) Wave functions are ( ) We will continue to focus on Schrödinger's representation to present the operators and relevant equations for the remaining examples.

Canonical Quantization
As a more complex example, we suggest the classical Hamiltonian, where x ∈  , given by subject to the restriction that ( ) 0 These two field operators should be self-adjoint operators when smeared with test functions. However, the positivity of φ means that π cannot be self adjoint. This situation complicates canonical quantization, and we do not discuss it Journal of High Energy Physics, Gravitation and Cosmology further. We will find that affine quantization is more friendly!

Affine Quantization
To proceed we introduce the classical affine field ( ) ( ) ( ) x φ > , and observe that the principal Poisson bracket is given by These classical fields are promoted to operators such that the principal commutator is subject to the condition that ( )

Affine Coherent States
The affine coherent states for this model are given by The fiducial vector is formally given by This relation leads to As before, and for a suitable factor K, the affine coherent states generate a resolution of unity such as and for the present set of coherent states, the Fubini-Study metric becomes Evidently, this affine metric is an infinite set of separate constant negative curvature spaces, specifically 2 ν −  , for every value of x. It is also noteworthy that the product of the coefficients of the two differential terms, i.e., coefficients of ( )

Enhanced Classical Operators
Following the single degree of freedom model, it follows that The enhanced classical Hamiltonian for this model is given by This relation "reduces" so that Finally, we find that The truly classical expression becomes with ( ) 0 x φ > , as expected.

Schrödinger's Representation and Equation
The Schrödinger representation for the principal operators is given by The Hilbert state vectors are given as functionals such as ( ) φ Ψ , and are formally normalized by ( ) Schrödinger's dynamical equation is now given by Regularization, by limiting the number of variables to a large but finite number of variables, enables such expressions to be investigated more clearly.

Quantum Gravity
The present paper is intended to be an introductory story to a more complete and higher level version of the study of quantum gravity that is available in [6].
That article offers more complex examples and the reader may wish to start with the more accessible story offered in the present article. In our present study, we emphasize the importance of choosing an affine quantization rather than a conventual canonical quantization. In so doing, we will not offer here a complete story of quantizing gravity but rather focus on some basic issues; to see a more complete story, the article [6] can be recommended.
One of the most difficult problems in quantum gravity deals with the choice of classical variables picked to promote to quantum operators. To briefly see that problem up close we first choose the classical phase space version of the gravitational action functional described in [7]. In particular, we introduce Instead, someone first changes the classical canonical coordinates, e.g.,   , and likely be self adjoint, but clearly the spectrum of the Hamiltonian operator would be different and thus the physics would be different.