For purposes of quantization, classical gravity is normally expressed by canonical variables, namely the metric
A single classical momentum p and coordinate q, with − ∞ < p , q < ∞ , and for which the Poisson bracket is { q , p } = 1 , are promoted to self-adjoint operators, p → P and q → Q , which satisfy [ Q , P ] = i ℏ . The canonical coherent states are given by | p , q 〉 ≡ exp ( − i q P / ℏ ) exp ( i p Q / ℏ ) | ω 〉 , where ( ω Q + i P ) | ω 〉 = 0 . Stationary variations of normalized vectors | ψ ( t ) 〉 of the quantum (q) action
A q = ∫ 0 T 〈 ψ ( t ) | [ i ℏ ( ∂ / ∂ t ) − H ( P , Q ) ] | ψ ( t ) 〉 d t (1)
lead to Schrödinger’s equation. The enhanced, with ℏ > 0 (as in the real world), classical (c) action is given by
A c = ∫ 0 T 〈 p ( t ) , q ( t ) | [ i ℏ ( ∂ / ∂ t ) − H ( P , Q ) ] | p ( t ) , q ( t ) 〉 d t = ∫ 0 T [ p ( t ) q ˙ ( t ) − H ( p ( t ) , q ( t ) ) ] d t . (2)
The connection of H with H is given by
H ( p , q ) = 〈 p , q | H ( P , Q ) | p , q 〉 = 〈 ω | H ( P + p , Q + q ) | ω 〉 = H ( p , q ) + O ( ℏ ; p , q ) , (3)
which, when ℏ → 0 , means H ( p , q ) = H ( p , q ) . Finally, we observe that the favored classical coordinates chosen to be promoted to operators, which, according to Dirac [
d σ c ( p , q ) 2 ≡ 2 ℏ [ ‖ d | p , q 〉 ‖ 2 − | 〈 p , q | d | p , q 〉 | 2 ] = ω − 1 d p 2 + ω d q 2 . (4)
A single classical momentum p and coordinate q, with − ∞ < p < ∞ but now 0 < q < ∞ , and for which the Poisson bracket is still { q , p } = 1 , are promoted to operators, p → P and q → Q , where 0 < Q < ∞ is self-adjoint, but P can not be self-adjoint, i.e., P † ≠ P . Instead of p, we choose to promote p q → ( P † Q + Q P ) / 2 ≡ D , a basic operator that is self-adjoint, i.e., D † = D , and which satisfies [ Q , D ] = i ℏ Q . The affine coherent states, chosen for simplicity with q and Q as dimensionless, are given by | p ; q 〉 ≡ exp ( i p Q / ℏ ) exp ( − i ln ( q ) D / ℏ ) | b 〉 , where [ ( Q − 1 ) + i D / b ℏ ] | b 〉 = 0 [
Stationary variations of normalized vectors | ψ ( t ) 〉 of the quantum action
A ′ q = ∫ 0 T 〈 ψ ( t ) | [ i ℏ ( ∂ / ∂ t ) − H ′ ( D , Q ) ] | ψ ( t ) 〉 d t (5)
lead to Schrödinger’s equation. The enhanced, with ℏ > 0 , classical action is given by
A ′ c = ∫ 0 T 〈 p ( t ) ; q ( t ) | [ i ℏ ( ∂ / ∂ t ) − H ′ ( D , Q ) ] | p ( t ) ; q ( t ) 〉 d t = ∫ 0 T [ − q ( t ) p ˙ ( t ) − H ′ ( p ( t ) q ( t ) , q ( t ) ) ] d t . (6)
The connection of H ′ with H ′ is now given by
H ′ ( p q , q ) = 〈 p ; q | H ′ ( D , Q ) | p ; q 〉 = 〈 b | H ′ ( D + p q Q , q Q ) | b 〉 = H ′ ( p q , q ) + O ′ ( ℏ ; p , q ) , (7)
which, when ℏ → 0 , means that H ′ ( p q , q ) = H ′ ( p q , q ) . Finally, we observe that
d σ a ( p , q ) 2 ≡ 2 ℏ [ ‖ d | p ; q 〉 ‖ 2 − | 〈 p ; q | d | p ; q 〉 | 2 ] = ( b ℏ ) − 1 q 2 d p 2 + ( b ℏ ) q − 2 d q 2 , (8)
which does not lead to Cartesian coordinates. Nevertheless, this metric is that of a constant negative curvature, whose value is − 2 / b ℏ ; see, e.g., [
The two versions of quantization described above apply to different problems. For example, assuming that − ∞ < p < ∞ , the Hamiltonian of the harmonic oscillator, e.g., H = ( p 2 + q 2 ) / 2 , and − ∞ < q < ∞ , canonical quantization works and affine quantization fails, while when 0 < q < ∞ , affine quantization works and canonical quantization fails [
The canonical story involves a constant zero curvature (i.e., a flat space), while the affine story involves a constant negative curvature, whose curvature value is − 2 / b ℏ . Such spaces are humanly visible only at one point, namely the “center point” where q = 1 [
The favorable Cartesian coordinates of canonical quantization are different from the favorable affine coordinates of affine quantization. However, extremely close to q = 1 the two versions are effectively the same. For a simple scalar field model, the metrics would be
d σ c 2 = ∫ [ A ( x ) − 1 d π ( x ) 2 + A ( x ) d φ ( x ) 2 ] d x (9)
for canonical, while that for affine is given by
d σ a 2 = ∫ [ B ( x ) − 1 φ ( x ) 2 d π ( x ) 2 + B ( x ) φ ( x ) − 2 d φ ( x ) 2 ] d x . (10)
These curves have the center-point curvature (where − 2 / B ( x ) denotes the value of the curvature). It is noteworthy that each point x has a visible center point where φ ( x ) = 1 , wherein its negative curvature at that x is given by − 2 / B ( x ) . Thus, if x is 1 dimensional, this leads to visible spots, or perhaps a line. If x = ( x 1 , x 2 ) is 2 dimensional, this leads to a visible set of those points, or a region where φ ( x 1 , x 2 ) = 1 with curvatures given by − 2 / B ( x 1 , x 2 ) , etc.
This expression captures the central point of the constant negative curvature where it appears nearly flat, which in fact is the only point that can be realized in our sight of the negative curvature. This behavior holds for each point x in space, thus enabling a field whose visible character is determined by the behavior of the non-dynamical term, B ( x ) .
Finally, observe that the canonical coherent states involve a single additional term of q in the expression (3), while, for the affine coherent states, the expression for q involves an exponential term in (7), passing from ln ( q ) to q, which entails a complete series. Thus, q enters as a product term due to D acting to “dilate” the result (hence the choice of symbol D). A similar exponential expression also emerges in the gravity coherent states; see [
The basic story of affine quantization, and how it applies to gravity, has been presented in [
Here we sketch an affine point of view for gravity. The classical momentric field, π b a ( x ) [ ≡ π a c ( x ) g b c ( x ) ] , and the metric field, g a b ( x ) , become the new basic variables, and these two variables have a joint set of Poisson brackets given by
{ 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. (11)
Passing to operator commutations, we are led by suitable coherent states [
[ 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. (12)
There are two irreducible representations of the metric tensor operator consistent with these commutations: one where the matrix { g ^ a b ( x ) } > 0 , which we accept, and one where the matrix { g ^ a b ( x ) } < 0 , which we reject.
The classical Hamiltonian for our models is given [
H ( π , g ) = ∫ { 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 ( 3 ) ( x ) } d 3 x , (13)
where R ( 3 ) ( x ) is the 3-dimensional Ricci scalar. For the quantum operators, we adopt a Schrödinger representation for the basic operators: specifically g ^ a b ( x ) = g a b ( x ) and
π ^ b a ( x ) = − 1 2 i ℏ [ g b c ( x ) ( δ / δ g a c ( x ) ) + ( δ / δ g a c ( x ) ) g b c ( x ) ] . (14)
It follows that the Schrödinger equation is given by
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 ( 3 ) ( x ) } d 3 x Ψ ( { g } , t ) , (15)
where { g } represents the g a b ( x ) matrix.
A Closer Look at the Affine Gravity MetricBased on the affine gravity coherent states [
d σ a ( π , g ) 2 = ∫ { ( b ( x ) ℏ ) − 1 g a b ( x ) g c d ( x ) d π b c ( x ) d π d a ( x ) + ( b ( x ) ℏ ) g a b ( x ) g c d ( x ) d g b c ( x ) d g d a ( x ) } d 3 x , (16)
which involves constant negative curvatures at every point x. To make that point more clearly, we can arrange to rephrase (16) as follows.
Let us introduce a strictly diagonal spatial metric given by
g [ ] ( x ) ≡ ( g [ 11 ] ( x ) 0 0 0 g [ 22 ] ( x ) 0 0 0 g [ 33 ] ( x ) ) , (17)
with a determinant given by det { g [ ] ( x ) } = g [ 11 ] ( x ) g [ 22 ] ( x ) g [ 33 ] ( x ) . Next, we introduce three orthogonal matrices given by
0 1 ( x ) ≡ ( 1 0 0 0 C 1 ( x ) − S 1 ( x ) 0 S 1 ( x ) C 1 ( x ) ) (18)
0 2 ( x ) ≡ ( C 2 ( x ) 0 − S 2 ( x ) 0 1 0 S 2 ( x ) 0 C 2 ( x ) ) (19)
0 3 ( x ) ≡ ( C 3 ( x ) − S 3 ( x ) 0 S 3 ( x ) C 3 ( x ) 0 0 0 1 ) (20)
where C a ( x ) ≡ cos ( θ a ( x ) ) , S a ( x ) ≡ sin ( θ a ( x ) ) , and 0 ≤ θ a ( x ) < 2 π . The connection now is given by
{ g a b ( x ) } = 0 1 ( x ) 0 2 ( x ) 0 3 ( x ) g [ ] ( x ) 0 3 ( x ) T 0 2 ( x ) T 0 1 ( x ) T , (21)
where T denotes transpose, as well as its complete interchange given by
g [ ] ( x ) ≡ 0 3 ( x ) T 0 2 ( x ) T 0 1 ( x ) T { g a b ( x ) } 0 1 ( x ) 0 2 ( x ) 0 3 ( x ) , (22)
exploiting the fact that 0 a ( x ) 0 a ( x ) T = 1 = 0 a ( x ) T 0 a ( x ) for each a.
This leads us to an alternative expression for (16) given by
d σ a ( π , g ) 2 ≡ ∫ { ( b ( x ) ℏ ) − 1 [ g [ a a ] ( x ) d π [ a a ] ( x ) ] 2 + ( b ( x ) ℏ ) [ g [ a a ] ( x ) d g [ a a ] ( x ) ] 2 } d 3 x . (23)
We have chosen a common factor b ( x ) and its inverse for all three terms. This is proper because the difference among each g [ a a ] ( x ) is simply their position along the matrix diagonal, which is a distinction with no physical significance.
Black holes are scattered around the universe and, loosely speaking, they tend to appear roughly similar. The appearance that pictures show is a cylindrical tube capped by a sprouting growth toward a flat topping; for sample pictures, see [
Equation (23) involves a wealth of potential constant negative curvatures as part of an affine gravity metric. They appear naturally different than Cartesian variables except at special points where g [ a a ] ( x ) = 1 = g [ a a ] ( x ) for appropriate x = ( x 1 , x 2 , x 3 ) . So far we have exploited orthogonal matrices in order to present a more orderly expression for d σ ( π , g ) a 2 . However, we have not yet explored changes of the underlying space parameterized by x. To consider a change of coordinates, we first suggest a new set of special coordinates, y = ( y 1 , y 2 , y 3 ) , that approximate a black hole in the equation
y 1 2 + y 2 2 − e y 3 = 1. (24)
This equation illustrates a cylindrical tube of approximate radius 1 when, e.g., − 6 < y 3 < 0 , but this tube blossoms out as y 3 passes through zero and rapidly requires an ever-expanding cylindrical radius of ( y 1 2 + y 2 2 ) 1 / 2 as y 3 > 0 2.
We next develop a connection between the x and the y variables. We seek to have g ˜ [ a a ] ( y ) = 1 = g ˜ [ a a ] ( y ) , for all [aa], when the three y coordinates satisfy (24). This requirement suggests adopting the three diagonal metric terms, i.e., g [ a a ] ( x ) and g ˜ [ a a ] ( y ) , and letting them act as “three-element vectors” such that g ˜ [ a a ] ( y ) = ( ∂ y a / ∂ x b ) g [ b b ] ( x ) , with a sum of b with [bb]. Additionally, we change π [ a a ] to obtain π ˜ [ a a ] ( y ) = ( ∂ x b / ∂ y a ) π [ b b ] ( x ) , with a sum of b with bb. Likewise, we introduce g ˜ [ a a ] ( y ) = ( ∂ x b / ∂ y a ) g [ b b ] ( x ) . It follows that
g ˜ [ a a ] ( y ) d g ˜ [ a a ] ( y ) = ( ∂ x b / ∂ y c ) ( ∂ y c / ∂ x d ) g [ d d ] ( x ) d g [ b b ] ( x ) = g [ b b ] ( x ) d g [ b b ] ( x ) , (25)
properly summed. A similar relation is
g ˜ [ a a ] ( y ) d π ˜ [ a a ] ( y ) = ( ∂ y b / ∂ x c ) ( ∂ x c / ∂ y d ) g [ d d ] ( x ) d π [ b b ] ( x ) = g [ b b ] ( x ) d π [ b b ] ( x ) , (26)
all of which leads to
d σ a ( π , g ) 2 ≡ ∫ { ( b ˜ ( y ) ℏ ) − 1 [ g ˜ [ a a ] ( y ) d π ˜ [ a a ] ( y ) ] 2 + ( b ˜ ( y ) ℏ ) [ g ˜ [ a a ] ( y ) d g ˜ [ a a ] ( y ) ] 2 } d 3 y . (27)
We next exam the expression b ˜ ( y ) ≡ b ( x ) for which the constant negative curvature is given by − 2 / b ˜ ( y ) ℏ . Assuming that one point of space, either denoted by x or by y, is the same as any other point of space, we are led to treat this term as similar, therefore, just as a constant, to be called b. Therefore, our final expression for the affine quantum metric is given by
d σ a ( π , g ) 2 ≡ ∫ { ( b ℏ ) − 1 [ g ˜ [ a a ] ( y ) d π ˜ [ a a ] ( y ) ] 2 + ( b ℏ ) [ g ˜ [ a a ] ( y ) d g ˜ [ a a ] ( y ) ] 2 } d 3 y . (28)
The choice of y for the background space has been so as to secure that the values of g ˜ [ a a ] ( y ) = 1 = g ˜ [ a a ] ( y ) when the three coordinate values obey the idealized black hole behavior y 1 2 + y 2 2 − e y 3 = 1 , or a portion thereof. If that is so, then the results for (28) have the appearance of being Cartesian coordinates, and thus may be seen as if they were actual Cartesian coordinates. Thus there would seem to be a sliver of nature that permits one to see starlight emitted along this “crack” in a black hole.
The quantization of Einstein’s version of classical gravity is possible using affine quantization instead of canonical quantization. Canonical quantization can not employ the promotion of classical Cartesian variables, as Dirac requires [
Thanks to Ewa Czuchry for suggesting several clarifications of the presentation.
The author declares no conflicts of interest regarding the publication of this paper.
Klauder, J.R. (2020) Quantum Gravity, Constant Negative Curvatures, and Black Holes. Journal of High Energy Physics, Gravitation and Cosmology, 6, 313-320. https://doi.org/10.4236/jhepgc.2020.63024