aa r X i v : . [ phy s i c s . g e n - ph ] M a y Geometric quantization rules in QCPB theory
Gen WANG ∗ School of Mathematical Sciences, Xiamen University,Xiamen, 361005, P.R.China
Abstract
Using the QCPB theory, we can accomplish the compatible combination of thequantum mechanics and general relativity supported by the G-dynamics. We furtherstudy the generalized quantum harmonic oscillator, such as geometric creation andannihilation operators, especially, the geometric quantization rules based on the QCPBtheory.
Contents ∗
1. WANG
This section mainly refers to the reference [1–6]. The quantum harmonic oscillator isdescribed by classical Hamiltonian operatorˆ H ( cl ) = b p ( cl )2 m + 12 mω x as one of the foundation problems of quantum mechanics, where b p ( cl ) = −√− ~ ddx , andpotential energy is V ( x ) = mω x in which ω is a constant frequency of the harmonicoscillator, in one dimension, the position of the particle was specified by a single coordinate, x . More precisely, it can be written as ˆ H ( cl ) = − ~ m d dx + mω x . The canonical commutationrelation between these operators is (cid:2) x, ˆ p ( cl ) (cid:3) QP B = √− ~ . In the ladder operator method,we define the ladder operators as follows a = r mω ~ (cid:18) x + √− mω b p ( cl ) (cid:19) , a † = r mω ~ (cid:18) x − √− mω b p ( cl ) (cid:19) The operator a is not Hermitian, since itself and its adjoint a † are not equal. Inversetransformation is x = r ~ mω (cid:0) a † + a (cid:1) , b p ( cl ) = √− r ~ mω (cid:0) a † − a (cid:1) And it says that the a and a † operators lower and raise the energy by ~ ω respectively. TheHamiltonian is simplified as ˆ H ( cl ) = ~ ω (cid:0) a † a + 1 / (cid:1) , where (cid:2) a, a † (cid:3) QP B = 1, in this ladderoperator method and the eigenvalues equation is given accordingly ˆ H ( cl ) | ψ i = E ( cl ) | ψ i , theharmonic oscillator eigenvalues or energy levels for the mode ω is E n ( cl ) = ~ ω ( n + 1 / , n = 0 , , , . . . Then the zero point energy follows E ( cl )0 = ~ ω/ n = 0, this paper, we denote E ( cl )0 ≡ E = ~ ω/
2. This form of the frequency is the same as that for the classical simple harmonicoscillator. The most surprising difference for the quantum case is the so-called zero-pointvibration of the n = 0 ground state.The general solution to the Schr¨odinger equation leads to a sequence of evenly spacedenergy levels characterized by a quantum number n . Let ξ = p mω ~ x , then the a and a † operators can be rewritten in respect to the ξ in the form a = 1 √ (cid:18) ξ + ddξ (cid:19) , a † = 1 √ (cid:18) ξ − ddξ (cid:19) In the quantum harmonic oscillator, one reinterprets the ladder operators as creation andannihilation operators, adding or subtracting fixed quanta of energy to the oscillator system.2. WANGCreation and annihilation operators are different for bosons (integer spin) and fermions(half-integer spin). This is because their wavefunctions have different symmetry properties.By using identities, ξφ n = r n φ n − + r n + 12 φ n +1 , ddξ φ n = r n φ n − − r n + 12 φ n +1 it leads to the results a | φ n i = √ n | φ n − i , a † | φ n i = √ n + 1 | φ n +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 an-nihilation operators. It also can be rewritten as a | n i = √ n | n − i , a † | n i = √ n + 1 | n + 1 i with the facts | n i = (cid:0) a † (cid:1) n √ n ! |i , h m | n i = δ mn , X n | n i h n | = 1where |i means the vacuum state. Furthermore, by using the number operator ˆ N = a † a , ithas h ˆ N , a i QP B = − a, h ˆ N , a † i QP B = a † (1)These two relations can be easily proven. As for creation and annihilation operators thatcan act on states of various types of particles, it also can refer to the ladder operators forthe quantum harmonic oscillator. In the latter case, the raising operator is interpreted asa creation operator, adding a quantum of energy to the oscillator system, similarly for thelowering operator. Mathematically, the creation and annihilation operators for bosons isthe same as for the ladder operators of the quantum harmonic oscillator. For instance, thecommutator of the creation and annihilation operators that are associated with the sameboson state equals one, while all other commutators vanish. However, for fermions themathematics is different, involving anticommutators instead of commutators. In this section, we begin with some basic concepts of quantum covariant Poisson bracketdefined by the generalized geometric commutator [7]. In quantum mechanics, quantumPoisson brackets is the commutator of two operators ˆ f , ˆ g defined as h ˆ f , ˆ g i QP B = ˆ f ˆ g − ˆ g ˆ f . Definition 1 (QCPB) . [7] The QCPB is generally defined as h ˆ f , ˆ g i = h ˆ f , ˆ g i QP B + G (cid:16) s, ˆ f , ˆ g (cid:17) in terms of quantum operator ˆ f , ˆ g , where G (cid:16) s, ˆ f , ˆ g (cid:17) = − G (cid:16) s, ˆ g, ˆ f (cid:17) is called quantumgeometric bracket.
3. WANGIt is zero if and only if ˆ f and ˆ g covariant commute, i,e. h ˆ f , ˆ g i = 0. It is remarkableto see that the QCPB representation admits a dynamical geometric bracket formula on themanifold. Note that structural function s represents the background property of spacetime.In the following discussions, we mainly propose and discuss an accurate expression of thequantum geobracket, maybe it’s one of situation in the general covariant form. Definition 2 (quantum geometric bracket) . [7] The quantum geometric bracket is G (cid:16) s, ˆ f , ˆ g (cid:17) = ˆ f [ s, ˆ g ] QP B − ˆ g h s, ˆ f i QP B where s represents the globally condition of space. As a consequence of the precise formulation of the quantum geometric bracket, the QCPBis completely written as h ˆ f , ˆ g i = h ˆ f , ˆ g i QP B + ˆ f [ s, ˆ g ] QP B − ˆ g h s, ˆ f i QP B
In this section, we will briefly review the entire theoretical framework of quantum covari-ant Hamiltonian system defined by the quantum covariant Poisson bracket completely givenby the previous paper [7].
Theorem 1 ( The covariant dynamics). [7] The covariant dynamics, the generalizedHeisenberg equation, G-dynamics can be formally formulated as
The covariant dynamics: D ˆ fdt = √− ~ h ˆ f , ˆ H i The generalized Heisenberg equation: d ˆ fdt = √− ~ h ˆ f , ˆ H i QP B − √− ~ ˆ H h s, ˆ f i QP B
G-dynamics: ˆ w = √− ~ h s, ˆ H i QP B .respectively, where D dt = ddt + ˆ w is covariant time operator, and ˆ H is the Hamiltonian and [ · , · ] denotes the GGC of two operators. Definitely, the QCPB theory leads us to a complete picture of the quantum mechanicsand its deep secrets. We can definitely employ the QCPB approach to discover more hiddenlaws of the quantum mechanics. More importantly, the QCPB theory can naturally reconcilethe quantum mechanics and the general relativity.Based on the G-dynamics, the imaginary geomenergy can be completely defined, we canbetter give a presentation for the covariant dynamics, ect.
Definition 3. [7] The imaginary geomenergy is defined by E (Im) ( ˆ w ) = √− ~ ˆ w , where ˆ w means G-dynamics.
4. WANGIt is obvious that the imaginary geomenergy can be expressed as E (Im) ( ˆ w ) = √− ~ ˆ w = h s, b H i QP B
Thus, a dynamical variable ˆ f with the geomenergy defined above, then covariant dynamicsis rewritten in the form √− ~ D dt ˆ f = h ˆ f , ˆ H i = √− ~ ddt ˆ f + ˆ f E (Im) ( ˆ w )As a consequence of the imaginary geomenergy, we can say that imaginary geomenergyis a new kind of Hamiltonian operator. Obviously, if the covariant equilibrium equation h ˆ f , ˆ H i = 0 holds, then it yields ddt ˆ f + ˆ f ˆ w = 0 or this covariant equilibrium formula inthe form shows ddt ˆ f = − ˆ f ˆ w , accordingly, ˆ f is said to be a quantum covariant conservedquantity. It’s convinced that the G-dynamics is a bridge to connect the quantum mechanicsand general relativity, it means the G-dynamics for the quantum gravity. This section mainly refers to the reference [7,8]. Commutator relations may look differentthan in the Schr¨odinger picture, because of the time dependence of operators. For example,consider the operators x ( t ), and ˆ p ( cl ) ( t ). The time evolution of those operators depends onthe Hamiltonian of the system. Considering the one-dimensional harmonic oscillator,ˆ H ( cl ) = ˆ p ( cl )2 m + mω x ddt x ( t ) = √− ~ [ ˆ H ( cl ) , x ( t )] QP B = ˆ p ( cl ) m (3) ddt ˆ p ( cl ) ( t ) = √− ~ [ ˆ H ( cl ) , ˆ p ( cl ) ( t )] QP B = − mω x As for the application of the QPB or the commutation, there are some many fields includingthe physics and mathematics, and so on. As a certainly example, we will now start by brieflyreviewing the quantum mechanics of a one-dimensional harmonic oscillator (2), and see howthe QCPB can be incorporated using GGC in the covariant quantization procedure. Withthe Hamiltonian given by (2), let’s use the QCPB to recalculate the (3), we can see howdifference emerges. More specifically, the covariant dynamics in terms of the position reads D dt x ( t ) = √− ~ h ˆ H ( cl ) , x ( t ) i = ˆ p ( cl ) ( t ) m + √− ~ x ( t ) ˆ p ( cl )2 s + 2ˆ p ( cl ) s ˆ p ( cl ) m By direct computation, the G-dynamics is given byˆ w ( cl ) = √− ~ ˆ p ( cl )2 s + 2ˆ p ( cl ) s ˆ p ( cl ) m (4)5. WANGAnd the generalized Heisenberg equation follows ddt x ( t ) = b p ( cl ) ( t ) m . In the same way, thecovariant dynamics for the momentum operator is D dt ˆ p ( cl ) ( t ) = √− ~ h ˆ H ( cl ) , ˆ p ( cl ) ( t ) i = − mω x − ˆ H ( cl ) u + b p ( cl ) ( t ) ˆ w ( cl ) where u = dsdx is used. Accordingly, the generalized Heisenberg equation appears ddt ˆ p ( cl ) ( t ) = − mω x − ˆ H ( cl ) u . Definition 4 ( Geomentum operator). [7] Let M be a smooth manifold represented bystructural function s , then geomentum operator is defined as ˆ p = −√− ~ D , where D = ∇ + ∇ s . The component is ˆ p j = −√− ~ D j in which D j = ∂ j + ∂ j s holds. Note that the geomentum operator is a revision of the classical momentum operator.
Theorem 2 (Geometric canonical quantization rules) . [7] Geometric equal-time canonicalcommutation relation is [ ˆ x i , ˆ p j ] = √− ~ D j x i , where [ · , · ] = [ · , · ] QP B + G ( s, · , · ) is QCPB. Geometric canonical commutation relation can be expressed in a specific form [ ˆ x i , ˆ p j ] = √− ~ (cid:16) δ ij + x i ∂∂x j s (cid:17) . In other words, it also can be rewritten as [ x i , ˆ p j ] = √− ~ θ ij , where θ ij = δ ij + x i u j , and u j = ∂ j s and ∂ j = ∂∂x j .The following geometric commutation relations can be easily obtained by substitutingthe geometric canonical commutation relation, Theorem 3.
The QCPB in terms of a, a † is (cid:2) a, a † (cid:3) = 1 + xu . This section mainly refers to the reference [8]. Denote by
Her the set of Hermitianoperators, and the set of non-Hermitian operators is denoted as
N Her , the set of skewHermitian operators are denoted by
SHer as well.In order to distinguish, let’s rewrite the geomentum operator as ˆ p = ˆ p ( ri ) = −√− ~ D from definition 4, it can be alternatively used in this paper. The covariant time evolutionof operators x ( t ) , ˆ p ( ri ) ( t ) depends on the Hamiltonian of the system. Considering the one-dimensional generalized quantum harmonic oscillator based upon geomentum operator 4,the Ri-operator as a Hamiltonian operator is given byˆ H ( ri ) = ˆ p ( ri )2 m + mω x T ( ri ) = ˆ p ( ri )2 m is the geometrinetic energy operator (GEO). The geomentum operatorin one-dimensional is b p ( ri ) = −√− ~ D dx , and D dx = d/dx + u . Accordingly, the covariantevolution of the position and geomentum operator based on the QCHS are respectivelygiven by: D dt x ( t ) = √− ~ [ ˆ H ( ri ) , x ( t )] (6)6. WANG D dt ˆ p ( ri ) ( t ) = √− ~ [ ˆ H ( ri ) , ˆ p ( ri ) ( t )]where [ · , · ] denotes the GGC of two operators. As a result, the covariant evolution of theposition is given by D dt x ( t ) = b p ( ri ) m + x b w ( ri ) , where G-dynamics is equal to b w ( ri ) = − √− ~ m (cid:18) u ddx + ddx u + 2 u (cid:19) (7)Similarly, the covariant evolution of the geomentum operator is D dt b p ( ri ) = − mω x − b H ( ri ) u − √− ~ b p ( ri ) h s, b H ( ri ) i QP B where the G-dynamics reads b w ( ri ) = − √− ~ h s, b H ( ri ) i QP B which is equivalent to the (7) as aprecise expression. Accordingly, the imaginary geomenergy follows E (Im) (cid:0) b w ( ri ) (cid:1) = √− ~ b w ( ri ) = ~ m (cid:18) u ddx + ddx u + 2 u (cid:19) The G-dynamics seen from the (7) can evidently say how it forms and works associated withthe structure function.
Consider the generalized quantum harmonic oscillator (5), the Ri-operator as a Hamil-tonian for it is precisely written as b H ( ri ) = − ~ m (cid:18) d dx + u (cid:19) + V ( x ) − E (Im) (cid:0) b w ( cl ) (cid:1) (8)where potential energy is V ( x ) = mω x , and the imaginary geomenergy is E (Im) (cid:0) b w ( cl ) (cid:1) = ~ m (cid:18) u ddx + ddx u (cid:19) Notice that the first three terms − ~ m (cid:16) d dx + u (cid:17) + V ( x ) are in a square form shown bythe Ri-operator while the unique terms E (Im) (cid:0) b w ( cl ) (cid:1) appears, and E (Im) (cid:0) b w ( cl ) (cid:1) = √− ~ b w ( cl ) is the imaginary geomenergy with respect to the G-dynamics b w ( cl ) . In fact, E (Im) (cid:0) b w ( cl ) (cid:1) implies that imaginary geomenergy has a unique status for the vacuum, it has nothing to dowith the matter, it’s a structural energy form, its energy spectrum represents the deepestsecrets of the space.The geometric canonical commutation relation [7] between these operator is (cid:2) x, b p ( ri ) (cid:3) = √− ~ + √− ~ xu
7. WANGIn the ladder operator method, we define the geometric creation operator and geometricannihilation operators a ( s ) = a + b, a ( s ) † = a † − b respectively, where b = q ~ mω u . By adirect computation, we can get the NG-operator given by a ( g ) = a − a † = q ~ mω ddx in terms ofthe annihilation operator a , similarly, a ( s )( g ) = a ( g ) + 2 b = q ~ mω D dx . As a consequence of thispoint, we get a ( s )( g ) − a ( g ) = 2 b = q ~ mω u = 0, as easily seen, the difference is only causedby geometric variable u induced by the derivative of the structure function s with respect tothe space. What’s more, we have a ( s ) † + a ( s ) = a † + a (9) a ( s ) † − a ( s ) = a † − a − b Inverse transformation is x = r ~ mω (cid:16) a ( s ) † + a ( s ) (cid:17) , b p ( ri ) = √− r ~ mω (cid:16) a ( s ) † − a ( s ) (cid:17) where b p ( ri ) = b p ( cl ) − √− ~ u ∈ N Her . With the support of (9), inverse transformation canbe rewritten as x = r ~ mω (cid:0) a † + a (cid:1) , b p ( ri ) = √− r ~ mω (cid:0) a † − a − b (cid:1) (10)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 a ( s ) † a ( s ) = mω ~ x − ~ mω D dx − / D dx = d dx + u + b w ( cl ) /b c , where b w ( cl ) /b c = 2 u ddx + ddx u (11)has been used and b c = − √− ~ m . More compactly, the geometric number operator b N ( r ) canbe simply expressed as b N ( r ) = b N + b N ( s ) , where b N ( r ) = a ( s ) † a ( s ) is denoted, where b N ( s ) = a † b − ba − b = − ~ mω b w ( cl ) /b c − ~ mω u b N = a † a = mω ~ x − ~ mω d dx − / b = ~ mω u , and a † b − ba = −√− b w ( cl ) /ω . Accordingly, it leads to the result b N ( s ) = − √− b w ( cl ) ω − ~ mω u
8. WANGThusly, the geometric number operator b N ( r ) can be rewritten in a clear form b N ( r ) = a † a − ~ mω u − √− b w ( cl ) ω (12)The Ri-operator (8) as a Hamiltonian is certainly expressed as b H ( ri ) = ~ ω (cid:16) b N ( r ) + 1 / (cid:17) (13)Plugging (12) into the (13), it yields b H ( ri ) = b H ( cl ) + b H ( s ) , where b H ( s ) = −√− ~ b w ( cl ) − ~ m u ,and E = ~ ω/ b H ( ri ) ψ = ~ ω (cid:16) b N + 1 / (cid:17) ψ + b H ( s ) ψ = E ( ri ) ψ where b H ( s ) ψ = ~ ω b N ( s ) ψ = E ( b ) ψ . The eigenvalues or energy levels of the Ri-operator is E n ( ri ) = ~ ω (cid:18) n + 12 (cid:19) + E ( b ) , n = 0 , , , . . . where E ( b ) = −√− ~ w ( q ) − ~ m u , and the energy levels are quantized at equally spacedvalues.As a result given by [8], we have obtained the following result √− ~ (cid:0) ∂ t − b w ( cl ) (cid:1) ψ = ˆ H ( ri ) ψ + E ( s ) ψ/ E ( s ) = ~ m u . Hence, inspired by the equation (14), it leads to the result b H ( ri ) + ~ m u = b H ( cl ) − E (Im) (cid:0) b w ( cl ) (cid:1) or in the form b H ( ri ) = ~ ω b N − ~ m u + b H ( hp ) , where b H ( cl ) = ~ ω b N + ~ ω/
2, and b H ( hp ) = ~ ω/ − √− ~ b w ( cl ) . Its eigenvalues can be given by E ( hp ) = E − √− ~ w ( q ) = ~ (cid:0) ω/ − √− w ( q ) (cid:1) (15)it implies that a connection between the E ( q ) = ~ w ( q ) and zero point energy E = ~ ω/ n = 0 corresponding to the vacuum condition. It reveals that the E ( q ) = ~ w ( q ) describes the vacuum structural energy. Then we get E ( hp ) / ~ = ω/ − √− w ( q ) . Inspiredby the zero point energy, it strongly implies that the √− w ( q ) has a special significance, asa geometric frequency, it naturally links to the algebra. Let’s consider the geometric creation operator a ( s ) and geometric annihilation operator a ( s ) † , a ( s ) = 1 √ (cid:18) ξ + D dξ (cid:19) , a ( s ) † = 1 √ (cid:18) ξ − D dξ (cid:19)
9. WANGwhere ξ = p mω ~ x . By a direct evaluation and comparative analysis, we have ddξ = χ ddx ,where χ = q ~ mω , then it yields D dξ = χ D dx , where dsdξ = χu . Accordingly, a ( s ) φ n = √ nφ n − + bφ n , a ( s ) † φ n = √ n + 1 φ n +1 − bφ n (16)As we can see, bφ n implies that it remains the numbers of particles, unlike the function ofthe creation and annihilation operators. a ( s ) | n i = √ n | n − i + b | n i a ( s ) † | n i = √ n + 1 | n + 1 i − b | n i where b = e χu = (cid:0) E ( s ) /E (cid:1) / /
2. In fact, by using the creation and annihilation operators,we can obtain the functions of both geometric creation and annihilation operators a ( s ) φ n = (cid:18) √ n + b √ n a † (cid:19) φ n − = (cid:18) bn a † (cid:19) aφ n a ( s ) † φ n = (cid:18) √ n + 1 − b √ n + 1 a (cid:19) φ n +1 = (cid:18) − bn + 1 a (cid:19) a † φ n where we have used φ n = a † φ n − √ n = aφ n +1 √ n +1 , as we can see, this is a new transformation for thecreation and annihilation operators, and weird one, and a = e χ − x + e χ ddx , a † = e χ − x − e χ ddx where e χ = χ √ , then b = e χu appeared previously, and then we can further get (cid:2) s, a † (cid:3) QP B = b, [ s, a ] QP B = − b . It leads to identity (cid:2) s, a † (cid:3) QP B + [ s, a ] QP B = 0. Hence, the geometriccreation operator and geometric annihilation operators can be rewritten as a ( s ) = a − [ s, a ] QP B , a ( s ) † = a † − (cid:2) s, a † (cid:3) QP B
Actually, (16) can be rewritten as a ( s ) φ n = aφ n + bφ n , a ( s ) † φ n = a † φ n − bφ n As mentioned previously, a † b − ba = −√− b w ( cl ) /ω , where b = e χu , we denote b N ( cg ) = a † b − ba = −√− b w ( cl ) /ω for a convenient discussions. Using the QCPB theory, we can do more calculations such as h ˆ N , a i = h ˆ N , a i QP B + G (cid:16) s, ˆ N , a (cid:17)
10. WANGand h ˆ N , a † i = h ˆ N , a † i QP B + G (cid:16) s, ˆ N , a † (cid:17) According to (1), we can go farther, by calculating the quantum geometric bracket G (cid:16) s, ˆ N , a (cid:17) and G (cid:16) s, ˆ N , a † (cid:17) , respectively. More precisely, G (cid:16) s, b N , a (cid:17) = b N [ s, a ] QP B − a h s, b N i QP B G (cid:16) s, b N , a † (cid:17) = b N (cid:2) s, a † (cid:3) QP B − a † h s, b N i QP B
By making use of the formula h s, b N i QP B = (cid:2) s, a † a (cid:3) QP B = (cid:2) s, a † (cid:3) QP B a + a † [ s, a ] QP B
Then quantum geometric bracket G (cid:16) s, ˆ N , a (cid:17) is precisely expressed as G (cid:16) s, ˆ N , a (cid:17) = ˆ N [ s, a ] QP B − a h s, ˆ N i QP B = ˆ N [ s, a ] QP B − a (cid:16)(cid:2) s, a † (cid:3) QP B a + a † [ s, a ] QP B (cid:17) = e χ (cid:16) a (cid:0) a † u − ua (cid:1) − ˆ N u (cid:17) = a b N ( cg ) − ˆ N b where we have used (cid:2) s, a † (cid:3) QP B = b, [ s, a ] QP B = − b . Meanwhile, quantum geometric bracket G (cid:16) s, ˆ N , a † (cid:17) precisely writes G (cid:16) s, b N , a † (cid:17) = b N (cid:2) s, a † (cid:3) QP B − a † h s, b N i QP B = b N (cid:2) s, a † (cid:3) QP B − a † (cid:16)(cid:2) s, a † (cid:3) QP B a + a † [ s, a ] QP B (cid:17) = e χ (cid:16) b N u + a † (cid:0) a † u − ua (cid:1)(cid:17) = a † b N ( cg ) + b N b where h s, ˆ N i QP B = √− w ( cl ) ω = − ˆ N ( cg ) and ˆ w ( cl ) = √− ω ˆ N ( cg ) . As a result, under the QCPB theory, we can get extensive resultsgiven by h b N , a i = − a + a b N ( cg ) − ˆ N b = a (cid:16) b N ( cg ) − b (cid:17) − ˆ N b h b
N , a † i = a † + a † b N ( cg ) + b N b = a † (cid:16) b N ( cg ) + ab + b (cid:17)
11. WANGwhere b b N ( cg ) = −√− b w ( cl ) /ω , plugging it into above geometriccommutations, then h ˆ N , a i = − a (cid:0) ˆ1 + √− w ( cl ) /ω (cid:1) − ˆ N b h ˆ N , a † i = a † (cid:0) ˆ1 − √− w ( cl ) /ω (cid:1) + ˆ N b
In conclusions, we obtain [ s, a ] QP B = − b (cid:2) s, a † (cid:3) QP B = b h s, ˆ N i QP B = √− w ( cl ) /ω More precisely, it can be shifted into the form [ s, a ] QP B = − b (cid:2) s, a † (cid:3) QP B = b h s, ˆ N i QP B = √− w ( q ) /ω By the way, we can get (cid:2) s, ˆ w ( cl ) (cid:3) QP B = [ s, − γ m (∆ s + 2 ∇ s · ∇ )] QP B = 2 γ m |∇ s | where [ s, ∇ s · ∇ ] QP B = −|∇ s | is easily deduced. In one dimensional case, it can be rewrittenas (cid:2) s, ˆ w ( cl ) (cid:3) QP B = 2 γ m u As a consequence, we further obtain h s, ˆ N ( cg ) i QP B = − (cid:20) s, h s, ˆ N i QP B (cid:21)
QP B = − √− ω (cid:2) s, ˆ w ( cl ) (cid:3) QP B = − √− γ m ω u Furthermore, let’s take the geometric number operator into account by using the QCPBtheory, then h b N ( r ) , a i = h b N + b N ( s ) , a i = h b N , a i + h b N ( s ) , a i (17) h b N ( r ) , a † i = h b N + b N ( s ) , a † i = h b N , a † i + h b N ( s ) , a † i Actually, we only need to evaluate the following QCPB, h b N ( s ) , a i = h b N ( s ) , a i QP B + G (cid:16) s, b N ( s ) , a (cid:17)
12. WANG h b N ( s ) , a † i = h b N ( s ) , a † i QP B + G (cid:16) s, b N ( s ) , a † (cid:17) where b N ( s ) = − √− b w ( cl ) ω − ~ mω u = a † b − ba − b or b N ( s ) = b N ( cg ) − b . In the next, by direct computations for two parts, it leads to h ˆ N ( s ) , a i QP B = h b N ( cg ) − b , a i QP B = h b N ( cg ) , a i QP B − (cid:2) b , a (cid:3) QP B and quantum geometric bracket follows G (cid:16) s, b N ( s ) , a (cid:17) = b N ( s ) [ s, a ] QP B − a h s, b N ( s ) i QP B = b − b N ( cg ) b − a h s, b N ( cg ) i QP B
Thusly, the QCPB in terms of b N ( s ) , a is equal to h b N ( s ) , a i = h b N ( s ) , a i QP B + G (cid:16) s, b N ( s ) , a (cid:17) = h b N ( cg ) , a i QP B − (cid:2) b , a (cid:3) QP B − b N ( cg ) b − a h s, b N ( cg ) i QP B + b Then, we get h ˆ N ( r ) , a i = h ˆ N , a i + h ˆ N ( s ) , a i = a (cid:18) b N ( cg ) − b − h s, b N ( cg ) i QP B (cid:19) − (cid:16) ˆ N + b N ( cg ) (cid:17) b + h ˆ N ( s ) , a i QP B + b Similarly, we have following result h ˆ N ( s ) , a † i QP B = h b N ( cg ) − b , a † i QP B = h b N ( cg ) , a † i QP B − (cid:2) b , a † (cid:3) QP B and G (cid:16) s, b N ( s ) , a † (cid:17) = b N ( s ) (cid:2) s, a † (cid:3) QP B − a † h s, b N ( s ) i QP B = b N ( cg ) b − a † h s, b N ( cg ) i QP B − b As a result, the QCPB in terms of b N ( s ) , a † is derived as h ˆ N ( s ) , a † i = h ˆ N ( s ) , a † i QP B + b N ( cg ) b − a † h s, b N ( cg ) i QP B − b