Loop Quantum Gravity is widely developed using canonical quantization in an effort to find the correct quantization for gravity. Affine quantization, which is like canonical quantization augmented and bounded in one orientation, e.g., a strictly positive coordinate. We open discussion using canonical and affine quantizations for two simple problems so each procedure can be understood. That analysis opens a modest treatment of quantum gravity gleaned from some typical features that exhibit the profound differences between aspects of seeking the quantum treatment of Einstein’s gravity.
We begin with two different quantization procedures, and two simple, but distinct, problems one of which is successful and the other one is a failure using both of the quantization procedures.
This exercise serves as a prelude to a valid and straightforward quantization of gravity.
The classical variables, p & q, that are elements of a constant zero curvature, better known as Cartesian variables, such as those featured by Dirac [
Our example is just the familiar harmonic oscillator, for which − ∞ < p , q < ∞ and a Poisson bracket { q , p } = 1 , chosen with a parameter-free, classical Hamiltonian, given by H ( p , q ) = ( p 2 + q 2 ) / 2 . The quantum Hamiltonian is H ( P , Q ) = ( P 2 + Q 2 ) / 2 , Schrödinger’s representation is given by P = − i ℏ ( ∂ / ∂ x ) and Q = x . Finally, for our example, Schrödinger’s equation is given by:
i ℏ ( ∂ ψ ( x , t ) / ∂ t ) = ( − ℏ 2 ( ∂ 2 / ∂ x 2 ) + x 2 ) / 2 ψ ( x , t ) . (1)
Solutions to Equation (1) for our example are well known. In particular, for the harmonic oscillator: eigenvalues are given by E n = ℏ ( n + 1 / 2 ) for n = 0 , 1 , 2 , ⋯ ; the eigenfunctions (with ℏ = 1 ) are given by ψ n ( x ) = N H n ( x ) e − x 2 / 2 with n = 0 , 1 , 2 , ⋯ ; H n ( x ) are the Hermite polynomials [
For our next example, we keep the same classical Hamiltonian, and we retain − ∞ < p < ∞ , but now we restrict 0 < q < ∞ . This new model is called a half-harmonic oscillator. It follows that the operator P ≠ P † , which leads to different behavior than when P was self-adjoint. In particular, we now have two distinct quantum Hamiltonians, specifically H 0 ( P , Q ) = ( P P † + Q 2 ) / 2 , while the other is H 1 ( P , Q ) = ( P † P + Q 2 ) / 2 . Each of these quantum Hamiltonians leads to the same classical Hamiltonian, namely, ( p 2 + q 2 ) / 2 , when ℏ → 0 . Hence, there are two valid sets of eigenvalues, specifically E 0 = ℏ [ ( 0,2,4, ⋯ ) + 1 / 2 ] and E 1 = ℏ [ ( 1,3,5, ⋯ ) + 1 / 2 ] . Moreover, these two eigenvalues can be weaved together, such as E = ℏ [ ( 0,2,3,5,7,8, ⋯ ) + 1 / 2 ] , which implies that there are infinitely many versions of the spectrum.
That judgement renders canonical quantization of the half-harmonic oscillator to be an invalid quantization.
The traditional classical affine variables are d ≡ p q and q > 0 , and they have a Poisson bracket given by { q , d } = q . Although the harmonic oscillator extends over the whole real line, we choose q to satisfy − b < q < ∞ , with b > 0 . For very large b we can approximate a full-line harmonic oscillator and even see what happens if we choose b → ∞ to mimic the full-line story.
The classical affine variables now are − ∞ < d ≡ p ( q + b ) < ∞ and 0 < ( q + b ) < ∞ , while the classical harmonic oscillator Hamiltonian is given by H ′ ( d , q ) = [ d 2 ( q + b ) − 2 + q 2 ] / 2 , an expression that obeys H ( p , q ) = ( p 2 + q 2 ) / 2 albeit that − b < q < ∞ .
Now we consider basic quantum operators, namely, D = [ P † ( Q + b ) + ( Q + b ) P ] / 2 and Q + b , which lead to [ Q + b , D ] = i ℏ ( Q + b ) , along with Q + b > 0 . The quantum partial-harmonic oscillator is now given by:
H ′ ( D , Q ) = [ D ( Q + b ) − 2 D + Q 2 ] / 2 , (2)
which, when b → ∞ , we find lim b → ∞ H ′ ( D , Q ) = ( P † P + Q 2 ) / 2 , for which P † = P now since − ∞ < Q < ∞ . Furthermore, lim b → ∞ [ Q + b , D + b P ] / b = lim b → ∞ [ Q , D + b P ] / b = [ Q , P ] = i ℏ 1 l .
In short, an affine quantization becomes a canonical quantization when the partial real line ( − b < q , Q < ∞ ) is stretched to its full length, ( − ∞ < q , Q < ∞ ).
Evidently, an affine quantization fails to quantize a full harmonic oscillator.
The common canonical operator expression, [ Q , P ] = i ℏ 1 l , directly implies [ Q , ( P Q + Q P ) / 2 ] = i ℏ Q , the basic affine operator! To confirm the affine expression, multiply i ℏ 1 l = [ Q , P ] by Q, which gives i ℏ Q = ( Q 2 P − P Q 2 ) / 2 = ( Q 2 P + Q P Q − Q P Q − P Q 2 ) / 2 , i.e., i ℏ Q = [ Q , ( Q P + P Q ) / 2 ] , which is the basic affine expression, [ Q , D ] = i ℏ Q , where D ≡ ( P Q + Q P ) / 2 . This derivation demands that Q > 0 and/or Q < 0 . Canonical quantization implies affine quantization but adds a limit on coordinates.2
Regarding our problem, now b = 0 , and so the classical affine variables are d ≡ p q and q > 0 , which lead to the half-harmonic oscillator H ′ ( d , q ) = ( d 2 q − 2 + q 2 ) / 2 . The basic affine quantum operators are D and Q > 0 , where ( P † Q + Q P ) / 2 ≡ D ( = D † ) and Q > 0 ( = Q † > 0 ) . These quantum variables lead to [ Q , D ] = i ℏ Q . The half-harmonic oscillator quantum Hamiltonian is given by H ′ ( D , Q ) = ( D Q − 2 D + Q 2 ) / 2 . Schrödinger’s representation is given by Q = x > 0 and,
D = − i ℏ [ x ( ∂ / ∂ x ) + ( ∂ / ∂ x ) x ] / 2 = − i ℏ [ x ( ∂ / ∂ x ) + 1 / 2 ] . (3)
Finally, Schrödinger’s equation is given by:
i ℏ ∂ ψ ( x , t ) / ∂ t = [ − ℏ 2 ( x ( ∂ / ∂ x ) + 1 / 2 ) x − 2 ( x ( ∂ / ∂ x ) + 1 / 2 ) + x 2 ] / 2 ψ ( x , t ) = [ − ℏ 2 ( ∂ 2 / ∂ x 2 ) + ( 3 / 4 ) ℏ 2 / x 2 + x 2 ] / 2 ψ ( x , t ) . (4)
Solutions of (4) have been provided by L. Gouba [
Finally, an affine quantization of the half-harmonic oscillator can be considered a correctly solved problem.
An important lesson from the foregoing set of examples is that canonical quantization requires classical variables, i.e., p & q, to be promoted to quantum operators must satisfy − ∞ < p , q < ∞ . However, an affine quantization requires classical variables, i.e., d & q, to be promoted to quantum operators must satisfy − ∞ < d < ∞ and − b < ± q < ∞ , provided that b is finite.
Our analysis of gravity will exploit the fact that the metric (the analog of q) is required to be strictly positive, i.e., d s ( x ) 2 = g a b ( x ) d x a d x b > 0 , provided { d x a } ≡ 0 . In addition, the metric itself, g a b ( x ) , is dimensionless. These important properties are well adapted to the possibilities offered by affine quantization.
To begin this section the author has chosen to offer the initial paragraph regarding “Loop Quantum Gravity” from Wikipedia in order to have a common text provided by proponents of that topic.3
“Beyond the Standard Model CMS Higgs-event.jpg Simulated Large Hadron Collider CMS particle detector data depicting a Higgs boson produced by colliding protons decaying into hadron jets and electrons Standard Model show Evidence show Theories show Supersymmetry hide Quantum gravity False vacuum String theory Spin foam Quantum foam Quantum geometry Loop quantum gravity Quantum cosmology Loop quantum cosmology Causal dynamical triangulation Causal fermion systems Causal sets Canonical quantum gravity Semiclassical gravity Superfluid vacuum theory show Experiments vte Loop quantum gravity (LQG) [
The story that follows may not correctly follow the history, but the author offers a reasonable attempt, especially regarding the most important features.
The basic elements for classical gravity are the metric g a b ( x ) = g b a ( x ) > 0 and the momentum π c d ( x ) = π d c ( x ) [
A partner expression A i a ( x ) is introduced to pair with E a i ( x ) , and these two variables lead to a Poisson bracket, effectively of unity, which is a key property for canonical quantization. The expression A i a ( x ) is not dimensionless, but, as canonical quantization requires, A i a ( x ) has the dimension of ℏ . Indeed, these expressions, when quantized, basically lead to [ E ^ a i ( x ) , A ^ j b ( x ′ ) ] = i ℏ δ j i δ a b δ 3 ( x , x ′ ) 1 l .
While we have begun the quantization study, it is essential to first ascertain whether or not we have found physically correct quantum operators or not. This issue arises because while there are many different classical variables that are acceptable for some expression, e.g., the pair of variables (skipping the usual indices) F ( E , A ) = F ¯ ( E ¯ , A ¯ ) , but, when quantized, F ^ ( E ^ , A ^ ) ≠ F ¯ ^ ( E ¯ ^ , A ¯ ^ ) . This fact
implies, generally, there is only one set of quantum operators that is physically correct, and some rules must decide which are the classical functions that promote physically correct quantum functions. For canonical quantization, Dirac [
The relevance of this paragraph is since the classical variables E a i ( x ) and A j b ( x ) are not Cartesian coordinates [
The present canonical gravity quantization leads to a very tiny, but not zero, discrete behavior for space and time [
The topic of quantum gravity has gathered numerous suggestions offered in Wikipedia [
In an affine quantization our favored variables are the metric g a b ( x ) , which is dimensionless, and the momentric π d c ( x ) ( ≡ π c e ( x ) g d e ( x ) ) , which has the dimension of ℏ .5 The Poisson brackets become:
{ π b a ( x ) , π d c ( y ) } = ( 1 / 2 ) [ δ d a π b c ( x ) − δ b c π d a ( x ) ] δ 3 ( x , y )
{ g a b ( x ) , π d c ( y ) } = ( 1 / 2 ) [ δ a c g b d ( x ) + δ b c g a d ( x ) ] δ 3 ( x , y )
{ g a b ( x ) , g c d ( y ) } = 0. (5)
In addition, we can promote these expressions to quantum operators, which then become:
[ π ^ b a ( x ) , π ^ d c ( y ) ] = i ( 1 / 2 ) ℏ [ δ d a π ^ b c ( x ) − δ b c π ^ a c ( x ) ] δ 3 ( x , y )
[ g ^ a b ( x ) , π ^ d c ( y ) ] = i ( 1 / 2 ) ℏ [ δ a c g ^ b d ( x ) + δ b c g ^ a d ( x ) ] δ 3 ( x , y )
[ g ^ a d ( x ) , g ^ c d ( y ) ] = 0. (6)
5Named for the momentum and the me tric.
Observe that the middle line of (5) and thus also that of (6), are still true if the metric variables change sign. This implies that using affine variables can lead to either strictly positive metrics, which we accept, or it could also lead to strictly negative metrics, which we reject. This important feature does not hold with analogous canonical classical and canonical quantum equations.
In the next section, we will establish the fact that the classical Poisson brackets in (5) match the quantum commutation relations in (6).
The basic affine quantum operators, g ^ a b ( x ) and π ^ d c ( x ) , are fundamental to the required coherent states, which are chosen [
| π ; g 〉 ≡ e ( i / ℏ ) ∫ π a b ( x ) g ^ a b ( x ) d 3 x e − ( i / ℏ ) ∫ η d c ( x ) π ^ c d ( x ) d 3 x | α 〉 , (7)
where { η ( x ) } is a 3 × 3 symmetric, arbitrary matrix, and { g ( x ) } ≡ e { η ( x ) } > 0 . The fiducial vector | α 〉 satisfies the equation [ ( g ^ a b ( x ) − δ a b 1 l ) + i π ^ d c ( x ) / α ( x ) ℏ ] | α 〉 = 0 , which establishes that 〈 α | g ^ a b ( x ) | α 〉 = δ a b and 〈 α | π ^ d c ( x ) | α 〉 = 0 .
For clarity, we offer a further description for a metric example given by:
( e w 1 ( x ) 0 0 0 e w 2 ( x ) 0 0 0 e w 3 ( x ) ) = exp ( w 1 ( x ) 0 0 0 w 2 ( x ) 0 0 0 w 3 ( x ) )
−1.5 em that we denote as W ( x ) = exp ( w ( x ) ) . Next, we introduce O ( x ) and its transpose O ( x ) T which are arbitrary, 3 × 3, orthogonal matrices such that O ( x ) T O ( x ) = O ( x ) O ( x ) T ≡ 1 l . A typical 3 × 3 orthogonal matrix, given as an example, is,
( 1 0 0 0 cos ( θ 1 ( x ) ) − sin ( θ 1 ( x ) ) 0 sin ( θ 1 ( x ) ) cos ( θ 1 ( x ) ) )
−1.5 em while changing 2 more positions of “1” leads to 3 angle parameters and to add 3 more parameters from W ( x ) , leads to 6 variables.
As expected, 6 is the number of parameters in g a b ( x ) = g b a ( x ) > 0 .
The Hamiltonian expressions for gravity are the most difficult part of the theory. According to ADM [
H ( π , g ) = ∫ { g − 1 / 2 ( x ) [ π b a ( x ) π a b ( x ) − 1 2 π a a ( x ) π b b ( x ) ] + g 1 / 2 ( x ) R ( x ) } d 3 x , (8)
where g ( x ) = det [ g a b ( x ) ] , and R ( x ) is the Ricci scalar term. Indeed, (8) can also be considered to be written directly in classical affine variables, namely g a b ( x ) and π d c ( x ) . In this case, we rename the canonical expression, H ( π , g ) , to an affine expression, H ′ ( π , g ) .
It is noteworthy that the momentum aspects of the classical Hamiltonian can be negative as well as positive. To simplify this aspect we adopt 2 × 2 metrics. In that case, the first factors of (8) could become π 1 1 = π 2 2 = 0 , π 2 1 = 1 , and π 1 2 = − 1 . In that case, π b a π a b = − 2 . For this example, the second factor of (8) π a a π b b = 0 . In addition, the Ricci scalar can be negative as well as positive. These properties carry over to the quantized elements, which could imply, a simple toy model, roughly like H = P 1 P 2 + Q 1 Q 2 .
The affine quantized version of expression (8) using metric, g ^ a b ( x ) , and momnentric operators, π ^ d c ( x ) , is given by:
H ′ ( π ^ , g ^ ) = ∫ { [ π ^ b a ( x ) g ^ − 1 / 2 ( x ) π ^ a b ( x ) − 1 2 π ^ a a ( x ) g ^ − 1 / 2 ( x ) π ^ b b ( x ) ] + g ^ 1 / 2 ( x ) R ^ ( x ) } d 3 x . (9)
Moreover, the connection between the classical and quantum terms arises from:
H ′ ( π , g ) = 〈 π ; g | H ′ ( π ^ , g ^ ) | π ; g 〉 = 〈 α | H ′ ( π ^ b a + π a c [ e η / 2 ] c e g ^ e d [ e η / 2 ] b d , [ e η / 2 ] c a g ^ a b [ e η / 2 ] d b ) | α 〉 , = H ′ ( 〈 α | π ^ b a | α 〉 + π a c [ e η / 2 ] c e 〈 α | g ^ e d | α 〉 [ e η / 2 ] b d , [ e η / 2 ] c a 〈 α | g ^ a b | α 〉 [ e η / 2 ] d b ) + O ( ℏ ; π , g ) , (10)
which becomes:
H ′ ( π b a , g c d ) = H ′ ( π b a , g c d ) + O ( ℏ ; π , g ) (11)
leading to the usual classical limit as ℏ → 0 , or when O is tiny enough to ignore. This expression confirms the equality of (5) and (6), as to the similarity of classical and quantum expressions.
In the previous section we developed the quantum Hamiltonian in (9), and now we develop a specific formulation wherein the dimensionless metric operator is given by g ^ a b ( x ) → g a b ( x ) and the dimensionless metric determinant is given by g ^ ( x ) ≡ det [ g ^ a b ( x ) ] → g ( x ) ≡ det [ g a b ( x ) ] , as well as the momentric, which has the dimensions of ℏ , and offers a formal expression given by:
π ^ d c ( x ) → − i ( 1 / 2 ) ℏ [ g d e ( x ) ( ∂ / ∂ g c e ( x ) ) + ( ∂ / ∂ g c e ( x ) ) g d e ( x ) ] . (12)
It is noteworthy that
π ^ b a ( x ) g ( x ) − 1 / 2 = 0, (13)
which is obtained by π ^ b a F ( g ) = 0 , and implies that [ g b c ( ∂ / ∂ g a c ) + 1 2 δ b a ] F ( g ) = 0 . Next, we find g b c g a c g d F ( g ) / d g + 1 2 δ b a F ( g ) = 0 , which requires that g d F ( g ) / d g + 1 2 F ( g ) = 0 ; hence F ( g ) ∝ g − 1 / 2 .
These expressions permit us to offer Schrödinger’s equation, which is,
i ℏ ( ∂ Ψ ( x , t ) / ∂ t ) = { ∫ 〚 π ^ b a ( x , t ) g ( x , t ) − 1 / 2 π ^ a b ( x , t ) − ( 1 / 2 ) π ^ a a ( x , t ) g ( x , t ) − 1 / 2 π ^ b b ( x , t ) + g ( x , t ) 1 / 2 R ( x , t ) 〛 d 3 x } Ψ ( x , t ) . (14)
Possible Solutions to Schröedinger EquationThe relation in (13) also leads to π ^ b a ( x ) Π y g ( y ) − 1 / 2 = 0 for any and all y. This property offers eigenfunctions having the general form Ψ ( x ) = V ( x ) Π y g ( y ) − 1 / 2 ; this expression is suggested here because in selected ultralocal procedures [
The author declares no conflicts of interest regarding the publication of this paper.
Klauder, J.R. (2021) Let Loop Quantum Gravity and Affine Quantum Gravity Examine Each Other. Journal of High Energy Physics, Gravitation and Cosmology, 7, 1027-1036. https://doi.org/10.4236/jhepgc.2021.73061