Geometric Potential Resulting from Dirac Quantization

TL;DR

This paper addresses the discrepancy in geometric potential formulations in Dirac quantization of particles on curved surfaces, proposing a scheme that yields the expected potential form by simultaneous quantization of key operators.

Contribution

It introduces a scheme for Dirac quantization that achieves the correct geometric potential by quantizing positions, momenta, and Hamiltonian simultaneously.

Findings

Resolves the discrepancy in geometric potential forms.

Identifies an operator-ordering-free quantization scheme.

Achieves the expected geometric potential form.

Abstract

A fundamental problem regarding the Dirac quantization of a free particle on an $N-1$ curved hypersurface embedded in $N$($\geq 2$) flat space is the impossibility to give the same form of the curvature-induced quantum potential, the geometric potential as commonly called, as that given by the Schr\"{o}dinger equation method where the particle moves in a region confined by a thin-layer sandwiching the surface. We resolve this problem by means of previously proposed scheme that hypothesizes a simultaneous quantization of positions, momenta, and Hamiltonian, among which the operator-ordering-free section is identified and is then found sufficient to lead to the expected form of geometric potential.

Equations

$${\chi }_1(x,p)\equiv f(x)\left(=0\right),\text{and }{\chi }_2(x,p)\equiv \mathbf{n}\cdot \mathbf{p}\left(=0\right).$$

$$[x_i,x_j{]}_D=0,[x_i,p_j{]}_D={\delta }_{ij}-n_i{n}_j,[p_i,p_j{]}_D=(n_j{n}_{i,k}-n_i{n}_{j,k})p_k,$$

$$\frac{d\mathbf{x}}{dt}\equiv [\mathbf{x},H{]}_D=\frac{\mathbf{p}}{\mu },\frac{d\mathbf{p}}{dt}\equiv [\mathbf{p},H{]}_D=-\frac{\mathbf{n}}{\mu }(\mathbf{p}\cdot \nabla n\cdot \mathbf{p}).$$

$$[f,g]=i\hslash O\left\{[f,g{]}_D\right\}$$

$$\mathbf{n}\cdot [\mathbf{x},H{]}_D=\mathbf{n}\cdot \frac{\mathbf{p}}{\mu }=0,\text{ and }\mathbf{n}\wedge [\mathbf{p},H{]}_D=0.$$

$$[x_i,x_j]=0,[x_i,p_j]=i\hslash \left({\delta }_{ij}-n_i{n}_j\right),[\mathbf{x},H]=i\hslash \frac{\mathbf{p}}{\mu },$$

$${\epsilon }_{[ij]}\left(n_i[p_j,H]+[p_j,H]n_i\right)=0,(i\ne j).$$

$$H=-\frac{{\hslash }^2}{2\mu }{\nabla }_{LB}^2+V_G,$$

$$\mathbf{p}=-i\hslash ({\nabla }_S+\frac{M\mathbf{n}}{2}).$$

$$[p_i,p_j]=\frac{i\hslash }{2}\left((n_j{n}_{i,l}-n_i{n}_{j,l})p_l+p_l(n_j{n}_{i,l}-n_i{n}_{j,l})\right),$$

$$H=\frac{p^2}{2\mu }-\frac{{\hslash }^2}{8\mu }{M}^2+V_G.$$

$$V_G=-\frac{{\hslash }^2}{4\mu }K+\frac{{\hslash }^2}{8\mu }{M}^2.$$

$$[p_j,H]=\frac{1}{2\mu }[p_j,p_k{p}_k]+[p_j,W]=\frac{1}{2\mu }\left([p_j,p_k]p_k+p_k[p_j,p_k]\right)+[p_j,W],$$

$$F_j$$

$$\stackrel{c.l.}{\Rightarrow }$$

$$G_j$$

$$\stackrel{c.l.}{\Rightarrow }$$

$${\epsilon }_{[ij]}\left(n_i{F}_j+F_j{n}_i\right)={\epsilon }_{[ij]}\left(n_i{G}_j+G_j{n}_i\right)=-\left(i/2\right)\hslash {\epsilon }_{[ij]}\left({\hslash }^2{n}_{i}K{,}_j\right).$$

$${\epsilon }_{[ij]}\left(n_i[p_j,W]+[p_j,W]n_i\right)=-2i\hslash {\epsilon }_{[ij]}{n}_{i}W{,}_j.$$

$${\epsilon }_{[ij]}\left(n_i[p_j,H]+[p_j,H]n_i\right)=-2i\hslash {\epsilon }_{[ij]}{n}_i\left(\frac{{\hslash }^2}{4\mu }K+W\right){,}_j=0.$$

$$-2i\hslash \mathbf{n}\wedge \mathbf{h}=0,$$

$$\frac{{\hslash }^2}{4\mu }K+W=\phi (x)f(x)+const.=const.I.e.W=const.-\frac{{\hslash }^2}{4\mu }K.$$

Full text

I Introduction

For a free particle constrained to live on a curved surface or a curved space, there is a curvature-induced potential in general, which however was born with some problems concerning its forms. DeWitt in 1957 used a specific generalization of Feynman’s time-sliced formula in Cartesian coordinates and found a surprising result that his amplitude turned out to satisfy a Schrödinger equation different from what had previously assumed by Schrödinger Sch and Podolsky podolsky . In addition to the kinetic energy which is Laplace-Beltrami operator divided by two times of mass, his Hamiltonian operator contained an extra effective potential proportional to the intrinsic curvature scalar. For a particle constrained on ($N-1$)-dimensional smooth curved surface $\Sigma^{N-1}$ in flat space $R^{N}$ ($N\succeq 2$), Jensen and Koppe in 1972 jk and subsequently da Costa dacosta in 1981 and 1982 developed the confining potential formalism (CPF) (also known as the thin-layer quantization) to deal with the free motion on the curved surface and demonstrated that the particle experiences a quantum potential that depends on both the square of the trace of, and the trace of the square of, the extrinsic curvature tensor of the curved surface, which was later called the geometric potential 2004 . By the CPF we mean to write the Schrödinger Equation within the uniform flat space within sufficiently high potential barriers on both sides of the surface, and then squeeze the width of barriers. Since the differnce between the excited and the ground state energy of the particle along the direction normal to the surface is very much larger than that of the particle along the tangential direction so that the degree of freedom along the normal direction is actually frozen to the ground state, an effective dynamics for the constrained system on the surface is thus established. This CPF has a distinct feature for no presence of any ambiguity. It is thus a powerful tool to examine various curvature-induced consequences in two-dimensional curved surfaces or curved wires packet0 . Experimental confirmations include: an optical realization of the geometric potential exp1 in 2010 and the geometric potential in a one-dimensional metallic $\mathit{C}_{60}$ polymer with an uneven periodic peanut-shaped structure in 2012 exp2 . Applying the CPF to momentum operators which are fundamentally defined as generators of a space translation, we have geometric momenta liu13-2 which depends on the extrinsic curvatures of the curved surface.

It is generally accepted that the canonical quantization offers a fundamental framework to directly construct the quantum operators, where the fundamental quantum conditions refer to a set of commutators between components of position and momentum dirac1 ; dirac2 . Many explorations have been devoted to searching for the *geometric potential *within the framework geoquan ; homma ; ogawa ; ikegami ; Jaffe ; mbbj ; ILNC ; kleinert ; Tagirov ; hong ; Golovnev ; 2011 ; weinberg . It is curious that no attempt is successful for even simplest two-dimensional curved surface $\Sigma^{2}$ embedded in $R^{3}$. Some results are contradictory with each other homma ; kleinert ; hong ; ogawa ; ikegami ; liu13-1 . We revisited all these attempts, and concluded that the canonical quantization together with Schrödinger-Podolsky-DeWitt approach of Hamiltonian operator construction was dubious, for the kinetic energy in it takes some presumed forms of distributing positions and momenta in the Hamiltonian. Since 2011, we have tried to enlarge the canonical quantization scheme to simultaneously quantize the Hamiltonian together with positions and momenta liu13-2 ; liu13-1 ; ECQ , rather than substitute the position and momentum operators into the presumed forms of Hamiltonian. Yet the success is limited because of the operator-ordering problem liu13-2 ; liu13-1 ; ECQ ; liu17 . It seems that the operator-ordering problem is inherent to the quantization and is hardly avoidable tdlee ; dirac3 . For instance, apart from the problem occurs in Hamiltonian, it even appears in the *fundamental quantum conditions, *see the last expression in (2). The key finding of the present study is to identify existence of the operator-ordering-free section in the enlarged scheme of quantum conditions, sufficiently to lead to the geometric potential.

II Dirac brackets and quantization conditions

Let us consider a non-relativistically free particle that is constrained to remain on a surface $\Sigma^{N-1}$ described by a constraint in the configurational space $f(x)=0$, where $f(x)$ is some smooth function of position $x$ in $R^{N}$, whose normal vector is $\mathbf{n}\equiv\nabla f(x)/|\nabla f(x)|$. We can always choose the equation of the surface such that $|\nabla f(x)|=1$ ikegami , so that $\mathbf{n}\equiv\nabla f(x)$. This is because no matter what form of the surface equation we begin with, only the unit normal vector and/or its derivatives enter the physics equation. In classical mechanics, the Hamiltonian is simply $H=p^{2}/2\mu$ where $p$ denotes the momentum, and $\mu$ denotes the mass. However, in quantum mechanics, we can not impose the usual canonical commutation relations $[x_{i},p_{j}]=i\hbar\delta_{ij}$, ($i,j=1,2,3,...N$). Dirac devised a general prescription to eliminate the motion in the direction normal to the surface by introducing the Dirac brackets dirac2 , $[f(x,p),g(x,p)]_{D}\equiv[f(x,p),g(x,p)]_{P}-[f(x,p),\chi_{\alpha}(x,p)]_{P}C_{\alpha\beta}^{-1}[\chi_{\beta}(x,p),g(x,p)]_{P}$, where $[f(x,p),g(x,p)]_{P}$ stands for the Poisson brackets, and repeated indices are summed over in whole of this Letter, and $C_{\alpha\beta}\equiv[\chi_{\alpha}(x,p),\chi_{\beta}(x,p)]_{P}$ are the matrix elements in the constraint matrix $\left(C_{\alpha\beta}\right)$ and the functions $\chi_{\alpha}(q,p)$ ($\alpha,\beta=1,2$) are two constraints weinberg ,

[TABLE]

It is an easy task to give following elementary Dirac brackets weinberg ,

[TABLE]

and weinberg ; liu17 ,

[TABLE]

The Dirac bracket quantization hypothesizes that the definition of a quantum commutator for any pair of variables $f$ and $g$ is given by dirac1 ; dirac2 ; dirac3 ,

[TABLE]

in which $O\left\{f\right\}$ stands for the quantum operator corresponding to the classical quantity $f$. The fundamental quantum conditions are widely taken to be comprised by following commutators $[x_{i},x_{j}],[x_{i},p_{j}]$ and $[p_{i},p_{j}]$. It must be mentioned that Dirac himself had never assumed so except when the particle moves in flat space where we should directly quantize the Poisson brackets, and he was clearly aware of the operator-ordering difficulties which should be carefully got over dirac3 . If taking the straightforward definition of quantum condition for different components of the momentum, $[p_{i},p_{j}]$, we encounter a disturbing operator-ordering problem in $O\left\{(n_{j}n_{i,k}-n_{i}n_{j,k})p_{k}\right\}$ homma ; ogawa ; ikegami . Much more annoying operator-ordering problem appears in $O\left\{\mathbf{n}(\mathbf{p\cdot\nabla n\cdot p})\right\}$ if one attempts to construct a quantum condition for $[\mathbf{p},H]$ liu17 .

However, in classical mechanics, we have following trivial consequences of (2) and (3), respectively,

[TABLE]

The first equation shows that the motion lies in the tangential plane, and the second shows that particle experiences no tangential force. The first equation is a single one, and the second one $\mathbf{n}\wedge[\mathbf{p},H]_{D}=0$ has $N(N-1)/2$ independent equations for its component form is $\varepsilon_{[ij]}n_{i}[p_{j},H]=0,(i\neq j)$ where $\varepsilon_{[ij]}\equiv\varepsilon_{i_{1}i_{2}...i...j...i_{N}}$ that is the Levi-Civita symbol of rank $N$, and the positions of two indexes $ij$ ($j\succ i$) in the array $i_{1}i_{2}...i...j...i_{N}$ of $\varepsilon_{[ij]}$ are arbitrary.

In order to transit to quantum mechanics for the system under study, the section of quantum conditions* free from the operator-ordering difficulty* is given by,

[TABLE]

The first finding of this Letter is that this set (6)-(7) is identified as the operator-ordering-free section of the enlarged set of quantum conditions defined by all commutators between $\{\mathbf{x},\mathbf{p},H\}$. To clearly demonstrate the critical importance of the ”trivial” relation (7), let us assume that the Hamiltonian operator is given by,

[TABLE]

where $V_{G}$ is the curvature-induced potential, and $\nabla_{LB}^{2}={\nabla_{S}}\cdot{\nabla_{S}}$ is the Laplace-Beltrami operator which is the dot product of the gradient operator ${\nabla_{S}\equiv}\mathbf{e}_{i}(\delta_{ij}-n_{i}n_{j})\partial_{j}$ $=\nabla_{N}-\mathbf{n}\partial_{n}$ on the surface $\Sigma^{N-1}$ with $\nabla_{N}$ being usual gradient operator in $R^{N}$ ikegami . The relation between the Laplace-Beltrami operator $\nabla_{LB}^{2}$ on $\Sigma^{N-1}$ and the usual Laplacian operator $\Delta_{N}\equiv\partial_{i}\partial_{i}$ in $R^{N}$ is $\nabla_{LB}^{2}={\nabla_{S}\cdot\nabla_{S}}$ $=\partial_{i}(\delta_{ij}-n_{i}n_{j})\partial_{j}=\Delta_{N}+M\partial_{n}-\partial_{n}^{2}$ with ${M}$ denoting the mean curvature that is in fact the trace of the extrinsic curvature tensor ikegami . It is straightforward to show that relations $\mathbf{p=}$ $[\mathbf{x},H](\mu/i\hbar)$ in (6) give the geometric momentum liu13-1 ,

[TABLE]

This geometric momentum is in fact the hermitian operator corresponding to $-i\hbar{\nabla_{S}\equiv}-i\hbar(\nabla_{N}-\mathbf{n}\partial_{n})$. In consequence, the commutators $[p_{i},p_{j}]$ turn out to satisfy the following relation with $f,_{k}\equiv\partial f/\partial x_{k}$,

[TABLE]

and the Hamiltonian operator turns out to be,

[TABLE]

The second and key finding of this Letter is: With the quantum condition (7) being imposed, $V_{G}$ must be the expected geometric potential dacosta ; 2004 ; packet0 ; ogawa ; ikegami ; Jaffe with $K\equiv\left(n_{i,j}\right)^{2}$ that is in fact the trace of square of the extrinsic curvature tensor ikegami ,

[TABLE]

*Proof: *To note that commutator $[p_{j},H]$ is,

[TABLE]

where $W\equiv V_{G}-{M}^{2}\hbar^{2}/(8\mu)$. Now we examine the expression $[p_{j},p_{k}]p_{k}+p_{k}[p_{j},p_{k}]\equiv F_{j}+G_{j}$, and substituting relation (10) into it, we have for $F_{j}$ and $G_{j}$ respectively,

[TABLE]

where $c.l.$ denotes the classical limit. Clearly, $F_{j}$ (14) vanishes in classical mechanics for $\mathbf{n\cdot}{\mathbf{p}}=0$, while $G_{j}$ (16) corresponds to the centripetal force $-2n_{j}{\mathbf{p}}\mathbf{\cdot}\nabla\mathbf{n\cdot}{\mathbf{p}}$. Thus, in the classical limit, the commutators $[\mathbf{p},H]$ is $-i\hbar\mathbf{n}\left(\mathbf{p\cdot\nabla n\cdot p}\right)/\mu$, but in general it never be $-i\hbar O\left\{\mathbf{n}\left(\mathbf{p\cdot\nabla n\cdot p}\right)\right\}/\mu$ because $[\mathbf{p},H]$ contains both the geometric potential that proportional to $\hbar^{2}$ and terms proportional to $\mathbf{n\cdot}{\mathbf{p}}$ that vanishes only in classical limit. In left-handed side of Eq. (7), we need to deal with $\varepsilon_{[ij]}\left(n_{i}F_{j}+F_{j}n_{i}\right)$ and $\varepsilon_{[ij]}\left(n_{i}G_{j}+G_{j}n_{i}\right)$, respectively. After somewhat lengthy but straightforward calculations, we find heavy cancellations among terms, and we find a very simple result,

[TABLE]

Next, we compute $\varepsilon_{[ij]}\left(n_{i}[p_{j},W]+[p_{j},W]n_{i}\right)$ in Eq. (7), which can be shown to be,

[TABLE]

The Eq. (7) is then,

[TABLE]

It is an orthogonal relation, which is in vector form,

[TABLE]

where $\mathbf{h}$ must be in parallel with normal $\mathbf{n}$ itself, and for convenience we assume $\mathbf{h}=\varphi(x)$ $\nabla f(x)$ where $\varphi(x)$ is an arbitrary function that can never be zero on any point of the surface $f(x)=0$ that immediately leads to relation $\nabla\left(\varphi(x)f(x)\right)=\varphi(x)\nabla f(x)$. In final, we obtain,

[TABLE]

Recalling $W\equiv V_{G}-{M}^{2}\hbar^{2}/(8\mu)$, we find that the geometric potential $V_{G}$ given by (22) differs from (12) by a constant which can be set to be zero. Q.E.D.

III Conclusions and discussions

The quantum conditions given by the straightforward applications of the quantization rule (4) are not always fruitful, even misleading. For the particle on the curved surface, in order to obtain the geometric potential predicted by the so-called CPF, a proper enlargement of the quantum conditions turns out to be compulsory to contain positions, momenta, and Hamiltonian. What is more, a construction of unambiguous quantum conditions out of the equation (4) proves inevitable. Combining the enlargement and the construction, we successfully obtain the geometric potential. Thus, the fundamental problem regarding the Dirac quantization of a free particle on a curved hypersurface is now resolved.

Finally, we would like to point out two points: 1, We do not know yet whether the operator-ordering-free section of the enlarged set of quantum conditions always exists in general, and if finding one, we will fix the Lee’s operator-ordering problem tdlee . 2, There are other forms of the enlargement and the construction of the quantum conditions in literature, for instance Refs. bender ; Deriglazov , but they took complete different forms and were devised to serve entirely different purposes.

Acknowledgements.

This work is financially supported by National Natural Science Foundation of China under Grant No. 11675051.