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arXiv:2005.11141v3 [physics.gen-ph] 5 Feb 2023

Geometric quantization rules in QCPB theory Gen WANG∗

Abstract Using the quantum covariant Poisson bracket (QCPB) theory, we can accomplish much more compatible explanations of the quantum mechanics supported by the G-dynamics. We further study the gen- eralized quantum harmonic oscillator equipped with the G-dynamics of type I, such as geometric creation and annihilation operators, and the geometric number operator as an extension of the number opera- tor is well-given for the deep discussions, the geometric Hamiltonian

operator is expressed as another form. Especially, the geometric quan- tization rules based on the QCPB theory is then calculated.

Contents 1 Introduction 2 1.1 Quantum harmonic oscillator . . . . . . . . . . . . . . . . . . 2 1.2 The quantum covariant Poisson bracket . . . . . . . . . . . . . 4 1.3 Covariant dynamics, generalized Heisenberg equation, G-dynamics 5 1.4 The QCPB for quantum harmonic oscillator . . . . . . . . . . 6 1.5 GEO for generalized quantum harmonic oscillator . . . . . . . 8

2 Generalized quantum harmonic oscillator 10 2.1 Ri-operator and geometric number operator . . . . . . . . . . 10

3 Geometric annihilation and creation operators 14 3.1 Geometric commutation . . . . . . . . . . . . . . . . . . . . . 17 3.1.1 Geometric commutation for number operator . . . . . . 17 3.1.2 Geometric commutation for geometric number operator 20 ∗ School of Mathematical Sciences, Xiamen University, Xiamen, 361005, P.R.China. email: wanggen@zjnu.edu.cn

4 Conclusions 23

1 Introduction 1.1 Quantum harmonic oscillator This section mainly refers to the reference [1, 2, 3, 4, 5, 6]. The quantum harmonic oscillator is described by classical Hamiltonian operator p̂(cl)2 1 Ĥ (cl) = + mω 2 x2 2m 2 as√one of the foundation problems of quantum mechanics, where p̂(cl) = d − −1~ dx , and potential energy is V (x) = 21 mω 2 x2 in which ω is a constant frequency of the harmonic oscillator, in one dimension, the position of the

particle was specified by a single coordinate x. More precisely, it can be written as ~2 d 2 1 Ĥ (cl) = − 2 + mω 2 x2 2m dx 2 The canonical commutation relation between these operators is x, p̂(cl) QP B = √ −1~. In the ladder operator method, we define the ladder operators as fol- lows r √ r √ mω −1 (cl) † mω −1 (cl) a= x+ p̂ , a = x− p̂

2~ mω 2~ mω The operator a is not Hermitian, since itself and its adjoint a† are not equal. Inverse transformation is r r ~ † (cl) √ ~mω † x= a + a , p̂ = −1 a −a 2mω 2 And it says that the a and a† operators lower and raise the energy by ~ω respectively. The Hamiltonian is simplified as Ĥ (cl) = ~ω a† a + 1/2

where a, a† QP B = 1, in this ladder operator method and the eigenvalues equation is given accordingly Ĥ (cl) |ψi = E (cl) |ψi, the harmonic oscillator eigenvalues or energy levels for the mode ω is En (cl) = ~ω (n + 1/2) , n = 0, 1, 2, . . .

(cl) Then the zero point energy follows E0 = ~ω/2 at n = 0, this paper, we (cl) denote E0 ≡ E0 = ~ω/2. This form of the frequency is the same as that for the classical simple harmonic oscillator. The most surprising difference for the quantum case is the so-called zero-point vibration of the n = 0 ground state. The general solution to the Schrödinger equation leads to a sequence of evenly p mω spaced energy levels characterized by a quantum number n. Let

† ξ= ~ x, then the a and a operators can be rewritten in respect to the ξ in the form 1 d † 1 d a= √ ξ+ , a = √ ξ− 2 dξ 2 dξ In the quantum harmonic oscillator, one reinterprets the ladder operators as annihilation and creation operators, adding or subtracting fixed quanta of energy to the oscillator system. Annihilation and creation operators are

different for bosons (integer spin) and fermions (half-integer spin). By using identities, r r r r n n+1 d n n+1 ξφn = φn−1 + φn+1 , φn = φn−1 − φn+1 2 2 dξ 2 2 it leads to the results √ √ a |φn i = n |φn−1 i , a† |φn i = n + 1 |φn+1 i

It is then evident that a† , in essence, appends a single quantum of energy to the oscillator, while a removes a quantum, this is the reason why they are referred to√as creation and annihilation √ operators. It also can be rewritten as a |ni = n |n − 1i , a† |ni = n + 1 |n + 1i with the facts n a† X |ni = √ | i , hm |ni = δmn , |ni hn| = 1 n! n

where | i means the vacuum state. Furthermore, by using the number oper- ator N̂ = a† a, it has h i h i N̂ , a = −a, N̂, a† = a† (1.1) QP B QP B

These two relations can be easily proven. As for annihilation and creation operators that can act on states of various types of particles, it also can

refer to the ladder operators for the quantum harmonic oscillator. In the latter case, the raising operator is interpreted as a creation operator, adding a quantum of energy to the oscillator system, similarly for the lowering op- erator. Mathematically, the annihilation and creation operators for bosons is the same as for the ladder operators of the quantum harmonic oscillator.

1.2 The quantum covariant Poisson bracket In this section, we begin with some basic concepts of quantum covariant Poisson bracket (QCPB) defined by the generalized geometric commutator [7]. In quantum mechanics, quantumh iPoisson bracket is the commutator ˆ ˆ of two operators f , ĝ defined as f , ĝ = fˆĝ − ĝ fˆ. The QCPB as an QP B extension of quantum Poisson bracket the is generally defined as h i h i

ˆ ˆ f , ĝ = f , ĝ ˆ + G s, f , ĝ QP B

in terms of quantum operator fˆ, ĝ, where h i G s, fˆ, ĝ = fˆ[s, ĝ]QP B − ĝ s, fˆ = −G s, ĝ, fˆ QP B

is called quantum geometric bracket, where the structural function or the geometric potential function s represents the globally condition of space. Hence, QCPB=QPB+quantum geometric bracket Note that the potential function here generally is a differentiable function. When the potential function is zero or constant, the quantum geometric brackets disappear. Clearly, zero and constant are special case. At this time, the quantum covariant Poisson brackets (QCPB) degenerate into quantum

Poisson brackets (QPB), and the corresponding results will degenerate into the results of classical quantum mechanics. Obviously, the extension of this quantum Poisson brackets is natural, mainly because the quantum theory after the extension has a wider adaptability, When s = 0 or s = constant, it degenerates to the form of classical quantum theory. h i It is zero if and only if fˆ and ĝ covariant commute, i,e. fˆ, ĝ = 0. It is remarkable to see that the QCPB representation admits a dynamical

geometric bracket formula on the manifold. Note that structural function s represents the background property of spacetime.

As a consequence of the precise formulation of the quantum geometric bracket, the QCPB is completely written as h i h i h i ˆ ˆ f , ĝ = f , ĝ + f [s, ĝ]QP B − ĝ s, fˆ ˆ QP B QP B

Note that the last two terms provided by quantum geometric bracket can be treated as the interactions fˆ, ĝ with the environments.

1.3 Covariant dynamics, generalized Heisenberg equa- tion, G-dynamics In this section, we will briefly review the entire theoretical framework of quantum covariant Hamiltonian system defined by the quantum covariant Poisson bracket completely given by the previous paper. More precisely, the time covariant evolution of any observable fˆ in the covariant dynamics is given by D fˆ 1 hˆ i =√ f , Ĥ dt −1~ It contains two independent dynamics:

The generalized h iHeisenberg equation h i of motion: dfˆ dt = √−1~ fˆ, Ĥ 1 − √−1~ Ĥ s, fˆ 1 . QP B h i QP B 1 The G-dynamics: ŵ = √−1~ s, Ĥ , QP B Note that since quantum operators generally h doinot satisfy commutativity, ˆ ˆ due to f ŵ 6= ŵf holds generally, it means fˆ, ŵ 6= 0, but here we for- QP B mally record dtD = dtd + ŵ to represent the quantum covariant time derivative, D ˆ and calculate strictly according to the derived result dt f = dtd fˆ + fˆŵ. The

specific calculation needs to be calculated according to the original definition of QCPB. Therefore, dt D = dtd + ŵ is still called covariant time operator, and Ĥ is the Hamiltonian and [·, ·] denotes the GGC of two operators. Note that the G-dynamics is actually a frequency operator for describing the frequency of the manifold space or the environment. Definitely, the QCPB theory leads us to a complete picture of the quan- tum mechanics and its deep secrets. We can definitely employ the QCPB

approach to discover more hidden laws of the quantum mechanics. More

importantly, the QCPB theory might reconcile the quantum mechanics and the gravity. Based on the G-dynamics, the imaginary geomenergy can be completely defined, we can better give a presentation for the covariant dynamics, ect. Definition 1. [7] Based on the G-dynamics h ŵ,i then the imaginary geomen- (Im) √ ergy is induced by E (ŵ) = −1~ŵ = s, Ĥ . QP B

It is obvious that the imaginary geomenergy can be expressed as √ h i E (Im) (ŵ) = −1~ŵ = s, Ĥ QP B

Thus, a dynamical variable fˆ with the geomenergy defined above, then co- variant dynamics is rewritten in the form √ D h i √ d −1~ fˆ = fˆ, Ĥ = −1~ fˆ + fˆE (Im) (ŵ) dt dt As a consequence of the imaginary geomenergy, we can say that imaginary geomenergy is a new kindh of Hamiltonian i operator. Obviously, if the covari- ˆ ant equilibrium equation f , Ĥ = 0 holds, then it yields dtd fˆ+ fˆŵ = 0 or this

covariant equilibrium formula in the form shows dtd fˆ = −fˆŵ, accordingly, fˆ is said to be a quantum covariant conserved quantity. It’s convinced that the G-dynamics is a bridge to deeply realize the quantum mechanics.

1.4 The QCPB for quantum harmonic oscillator This section mainly refers to the reference [7, 8]. Commutator relations may look different than in the Schrödinger picture, because of the time dependence of operators. For example, consider the operators x(t), and p̂(cl) (t). The time evolution of those operators depends on the Hamiltonian of the system. Considering the one-dimensional harmonic oscillator,

p̂(cl)2 mω 2 x2 Ĥ (cl) = + (1.2) 2m 2 the evolution of the position and momentum operators is given by: √ d −1 (cl) p̂(cl) x(t) = [Ĥ , x(t)]QP B = (1.3) dt ~ m 6

√ d (cl) −1 (cl) (cl) p̂ (t) = [Ĥ , p̂ (t)]QP B = −mω 2 x dt ~ As for the application of the QPB or the commutation, there are some many fields including the physics and mathematics, and so on. As a certainly example, we will now start by briefly reviewing the quantum mechanics of a one-dimensional harmonic oscillator (1.2), and see how the QCPB can be incorporated using GGC in the covariant quantization procedure. With the

Hamiltonian given by (1.2), let’s use the QCPB to recalculate the (1.3), we can see how difference emerges. More specifically, the covariant dynamics in terms of the position reads √ D −1 h (cl) i p̂(cl) (t) x (t) = Ĥ , x (t) = + xŵ (cl) dt ~ m

By direct computation, the G-dynamics in terms of Ĥ (cl) is given by h i √ √ −1 p̂(cl)2 s + 2p̂(cl) sp̂(cl) ŵ (cl) = − −1 s, Ĥ (cl) /~ = (1.4) QP B ~ 2m d = bc 2u + ux dx √ 2 −1~ ds d where bc = − 2m , and u = dx , ux = dx 2 s are used. Note that the G- (cl) dynamics in terms of Ĥ describes the quantum rotation of the manifolds

space in a certainty quantum system, its eigenvalues represent the frequency spectrum. Furthermore, it gets √ h i −1~ŵ (cl) = s, Ĥ (cl) QP B

(cl) And the generalized Heisenberg equation follows dtd x (t) = p̂ m(t) . In the same way, the covariant dynamics for the momentum operator is √ D (cl) −1 h (cl) (cl) i p̂ (t) = Ĥ , p̂ (t) = −mω 2 x − Ĥ (cl) u + p̂(cl) (t) ŵ (cl) dt ~ ds where u = dx is used. Accordingly, the generalized Heisenberg equation appears d (cl) p̂ (t) = −mω 2 x − Ĥ (cl) u dt

Definition 2 (Geomentum operator). [7] Let M be a smooth manifold represented by structural function s, then geomentum operator is defined as √ p̂ = − −1~D √ where D = ∇+∇s. The component is p̂j = − −1~Dj in which Dj = ∂j +∂j s holds.

Note that the geomentum operator is a revision of the classical momentum operator.

Theorem 1 (Geometric canonical quantization rules). [7] Geometric equal- time canonical commutation relation is √ [xi , pˆj ] = −1~Dj xi

where [·, ·] = [·, ·]QP B + G (s, ·, ·) is QCPB.

Geometric canonical commutation relation can be expressed in a specific form √ [xi , pˆj ] = −1~ (δij + xi uj ) √ In other words, it also can be rewritten as [xi , pˆj ] = −1~θij , where θij = δij + xi uj , and uj = ∂j s and ∂j = ∂x∂ j .

1.5 GEO for generalized quantum harmonic oscillator This section mainly refers to the reference [8]. In√order to distinguish, let’s rewrite the geomentum operator as p̂ = p̂(ri) = − −1~D from definition 2, it can be alternatively used in this paper. The covariant time evolution of operators x(t), p̂(ri) (t) depends on the Hamiltonian of the system. Considering the one-dimensional generalized quantum harmonic oscilla- tor based upon geomentum operator 2, the Ri-operator as a Hamiltonian

p̂(ri)2 mω 2 x2 E (s) √ Ĥ (ri) = + = Ĥ (cl) − − −1~ŵ (cl) (1.5) 2m 2 2 where p̂(ri)2 √ T̂ (ri) = = T̂ (cl) − E (s) /2 − −1~ŵ (cl) 2m

is the geometrinetic energy operator (GEO), where √ ~2 d d The imaginary geomenergy here: E (Im) ŵ (cl) = −1~ŵ (cl) = 2m 2u dx + dx u , ~2 d2 Classical kinetic energy operator: T̂ (cl) = − 2m dx2 , (s) ~2 2 The geometric potential energy function: E = m u . √ D D The geomentum operator in one-dimensional is p̂(ri) = − −1~ dx , and dx = d/dx+u. Accordingly, the covariant evolution of the position and geomentum

operator based on the QCHS are respectively given by: √ D −1 (ri) x(t) = [Ĥ , x(t)] (1.6) dt ~ √ D (ri) −1 (ri) (ri) p̂ (t) = [Ĥ , p̂ (t)] dt ~ where [·, ·] denotes the GGC of two operators. As a result, the covariant evolution of the position is given by

D p̂(ri) x (t) = + xŵ (ri) dt m where G-dynamics is equal to √ (ri) −1~ d d 2 ŵ = − 2u + u + 2u (1.7) 2m dx dx

And the generalized Heisenberg equation with respect to x follows √ h i d −1 (ri) p̂(ri) (t) x (t) = Ĥ , x (t) = dt ~ QP B m Similarly, the covariant evolution of the geomentum operator is √ D (ri) 2 (ri) −1 (ri) h i p̂ = −mω x − Ĥ u − p̂ s, Ĥ (ri) dt ~ QP B

The generalized Heisenberg equation with respect to the geomentum operator p̂(ri) becomes d (ri) p̂ (t) = −mω 2 x − Ĥ (ri) u dt

√ h i And the G-dynamics reads ŵ (ri) = − ~−1 s, Ĥ (ri) which is equivalent QP B to the (1.7) as a precise expression. Accordingly, the imaginary geomenergy follows (Im) (ri) √ (ri) ~2 d d 2 E ŵ = −1~ŵ = 2u + u + 2u 2m dx dx The G-dynamics seen from the (1.7) can evidently say how it forms and works associated with the geometric potential function.

2 Generalized quantum harmonic oscillator 2.1 Ri-operator and geometric number operator Consider the generalized quantum harmonic oscillator (1.5), the Ri-operator as a Hamiltonian for it is precisely written as 2 (ri) ~2 d 2 √ Ĥ =− + u + V (x) − −1~ŵ (cl) (2.1) 2m dx2 where: The potential energy is V (x) = 12 mω 2 x2 , √ The imaginary geomenergy related to the type I is E (Im) ŵ (cl) = −1~ŵ (cl) .

~2 d2 The one-dimensional classical kinetic energy operator: T̂ (cl) = − 2m dx2 , (s) ~2 2 The geometric potential energy function: E = m u . ~ 2 d 2 Notice that the first three terms − 2m dx2 + u2 + V (x) are generally in a square form shown by the Ri-operator while the unique terms–the imag- (Im) (cl) inary geomenergy with respect to the G-dynamics of type I E ŵ = √ (cl) (Im) (cl)

−1~ŵ appears. In fact, E ŵ implies that imaginary geomenergy has a unique status, it has nothing to do with the matter, it’s a structural energy form, its energy spectrum represents the deepest secrets of the space. The geometric canonical commutation relation [7] between these operator is (ri) √ √ x, p̂ = −1~ + −1~xu In the ladder operator method, we define the geometric annihilation operator

and geometric creation operators † a(s) = a + b, a(s) = a† − b

q ~ respectively, where b = 2mω u is directly associated with the line curvature u 6= 0. By a direct q computation, we can get the NG-operator given by 2~ d a(g) = a − a† = mω dx in terms of the annihilation operator a, similarly, q 2~ D a(s)(g) = a(g) + 2b = mω dx . As a consequence of this point, we get r 2~ a(s)(g) − a(g) = 2b = u 6= 0 mω As easily seen, the difference is only caused by geometric variable u induced by the derivative of the structure function s with respect to the space. What’s

more, we have † a(s) + a(s) = a† + a (2.2) † a(s) − a(s) = a† − a − 2b

Inverse transformation is r r ~ (s) † (s) (ri) √ ~mω (s) † x= a +a , p̂ = −1 a − a(s) 2mω 2 where √ p̂(ri) = p̂(cl) − −1~u = p̂(cl) − s, p̂(cl) (t) QP B clearly is a non-Hermitian operator. With the support of (2.2), inverse trans- formation can be rewritten as r r

~ √ ~mω † x= a† + a , p̂(ri) = −1 a − a − 2b (2.3) 2mω 2 in terms of the a and a† operators. Thus, we can then show that a general result for a and a† operators is given by

† mω 2 ~ D2 a(s) a(s) = x − − 1/2 2~ 2mω dx2 2 2 D d 2 (cl) where dx 2 = dx2 + u + ŵ /bc , where

d du ŵ (cl) /bc = 2u + (2.4) dx dx

√ −1~ has been used and bc = − 2m . More compactly, the geometric number (r) operator N̂ can be simply expressed as † N̂ (r) = a(s) a(s) = N̂ + N̂ (s)

It’s obvious that this is a natural generalization of the number operator, the term N̂ (s) nicely remedies the shortage of the number operator, it helps to broaden the quantum range wider to study. More precisely, a(s)† a(s) = a† − b (a + b) = N̂ + N̂ (s) = a† a + a† b − ba − b2

where ~ ~ 2 N̂ (s) = a† b − ba − b2 = −ŵ (cl) /bc − u 2mω 2mω mω 2 ~ d2 N̂ = a† a = x − − 1/2 2~ 2mω dx2 where b2 = 2mω ~ u2 , and mixed term is expressed as

~ √ N̂ (mix) (a, a† , b) = a† b − ba = − ŵ (cl) /bc = − −1ŵ (cl) /ω 2mω that is only linked to the G-dynamics of the type I, conversely, the term a† a − b2 can be regarded as the pure term. In fact, the mixed term implies the essence of the G-dynamics which exists as a frequency operator for the frequency spectrum. The details for a† a can be given by † mω 2 p̂(cl)2 ~ mω 2 p̂(cl)2

aa= x + − = x + − 1/2 2~ (mω)2 mω 2~ 2~mω

Accordingly, it leads to the result √ (s) −1ŵ (cl) ~ 2 N̂ = − − u ω 2mω

Thusly, the geometric number operator N̂ (r) can be rewritten in a clear form

~ 2 √ ŵ (cl) N̂ (r) = a† a − u − −1 (2.5) 2mω ω 12

The Ri-operator (2.1) as a Hamiltonian is certainly expressed as (ri) (r) Ĥ = ~ω N̂ + 1/2 (2.6)

Plugging (2.5) into the (2.6), it yields Ĥ (ri) = ~ω a(s)† a(s) + 1/2 = ~ω a† a − b2 + a† b − ba + 1/2 √ † −1ŵ (cl) ~ 2 = ~ω a a − − u + 1/2 ω 2mω √ † −1ŵ (cl) ~ 2 = ~ω a a + 1/2 − ~ω + u ω 2mω † √ (cl

Continued Discussion and Extended Analysis