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A note on the quantum Hamiltonian for a free particle
Volume 156, number 1,2
PHYSICS LETTERS A
3 June 1991
A note on the quantum Hamiltonian for a free particle I.M. B e n n Department of Mathematics, Universityof Newcastle, Newcastle, New South Wales,Australia
and R.W. Tucker School of Physics and Materials, Universityof Lancaster, LancasterLA1 4YB, UK
Received 22 March 1991; accepted for publication 8 April 199 I Communicated by J.P. Vigier
We reconsider the non-relativistic quantum mechanics of a free particle on a Riemannian manifold and offer an argument for an invariant quantum Hamiltonian proportional to the Laplacian amended by one quarter of the scalar curvature oftbe manifold. An explicit ordering of the arbitrary canonical variables in the classical Hamiltonian is found that implies the existence of a quantum potential that can be made to vanish at any point by adopting geodesic coordinates for that region of the manifold.
1. Introduction It is a sobering thought that there is no clear consensus as to the correct quantum description of a nonrelativistic free particle moving on a Riemannian manifold. Even for a particle in R 3 there is no known unambiguous transcription from the classical to the q u a n t u m Hamiltonian. The Schr6dinger representation o f the canonical c o m m u t a t i o n relations leads to operator ordering ambiguities, and the q u a n t u m Hamiltonian obtained depends u p o n the coordinates in which the transcription takes place. However, in this R 3 case it is generally agreed that the transcription should result in the "Laplacian" on normalisable state functions as Hamiltonian operator, and ad hoe prescriptions can be given to ensure that this is the case when quantising in other than Cartesian coordinates [ 1-3 ]. The above ambiguities are far more troublesome for the canonical quantisation o f a free classical particle moving on a geodesic o f an arbitrary Riemannian manifold. In this case we cannot simply introduce extra prescriptions to ensure that the "correct" q u a n t u m Hamiltonian is obtained in that it is far less clear what the "correct" Hamiltonian is. Specifically
one can add curvature scalar terms to the Laplacian. Indeed it has been widely argued that the appropriate Hamiltonian operator differs from the Laplacian by the addition of~R, where R is the curvature scalar [4,5]. Ultimately the arguments for this curvature term hinge on obtaining the "expected" wave equation in "special" coordinates. These special coordinates {ya} are chosen such that at their origin the connection coefficients Fabc vanish, with the derivatives satisfying O F~/+~O F '% O -Oy -~ Oy ~ ca b-fTy~Fab ~= 0 . Although such coordinates are often referred to as normal we prefer to reserve this terminology for geodesic coordinates. These special coordinates are not geodesic coordinates, although they are closely related to them [6 ]. If the canonical commutation relations are realised through the Schr'6dinger representation then the ambiguities inherent in the transcription from classical to q u a n t u m Hamiltonian can be resolved by requiting that we not only transcribe the classical coordinates and m o m e n t a to their q u a n t u m counterparts, but in addition add a " q u a n t u m po-
0375-9601/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)
1
V o l u m e 156, n u m b e r 1,2
PHYSICS LETTERS A
tential" to the classical Hamiltonian [ 7 ]. This quantum potential will depend upon the coordinates in which the transcription takes place as well as upon the ordering of the variables in the classical Hamiltonian. Of course the ambiguity now becomes the choice of quantum potential. For example, we can essentially declare the quantum Hamiltonian operator to be the Laplacian and then choose the quantum potential to be the difference between the Laplacian and what is obtained by replacing classical p's and q's with their quantum counterparts. The quantum potential will then involve the connection coefficients, for the coordinate basis in which the transcription takes place, and their derivatives. We shall show that there is a unique ordering of the classical variables so that the SchrSdinger substitution for the p's and q's may be followed by the addition of a quantum potential involving the connection coefficients but not their derivatives to result in an invariant Hermitian quantum Hamiltonian. This Hamiltonian differs from the Laplacian by a ~R term.
2. Wave functions and densities
A direct comparison of papers on this subject is inconvenienced by the fact that some authors represent the state of the particle by a complex function on the configuration manifold, independent of any coordinate system, whereas others represent it by a complex half-density which, upon a coordinate transformation, changes by the square root of the determinant of the transformation matrix. For the case of an orientable Riemannian manifold these two approaches are equivalent as, to facilitate comparisons, we now show. Let M be an n-dimensional orientable Riemannian manifold with metric tensor g and orienting volume form • 1. Then a Hermitian inner product on complex functions ~ and ~u is defined by (¢, ~,) = f 0 " ~ , 1 .
(1)
M
The volume n-form • 1 can be written in coordinates {M} as • 1 =o'dx
where
I ,
(2)
3 J u n e 1991
a= Ideggol ~/2
(3)
and d x I = d x 1 AOx2A ...A d x n. If O=a'/2(~ then ~ is a half-density. A Hermitian inner product may be defined on half-densities by (~, ~ ) = j ~q)dx ~
(4)
M
so that
(¢, ~) = (~, ~ ) .
(5)
If 0 is any operator taking functions to functions then an operator taking half-densities to half-densities is given by
O=al/20a-l/2.
(6)
This relation between operators gives an isomorphism of the algebra of operators in that ( O ( 02 ) = 0102. It also readily follows that 0 is selfadjoint with respect to ( , ) if and only if 0 is selfadjoint with respect to ( , ). Thus, for this case of an orientable Riemannian manifold, any quantum description involving halfdensities has a parallel description in terms of functions. In the following we shall use the latter description and assume M compact to facilitate the definition of Hermitian operators.
3. The quantum mechanics of a free particle
An arbitrary vector field X is prevented from being anti-self-adjoint with respect to the inner product ( 1 ) due to a term involving the Lie derivative of the volume form. We can obtain an anti-self-adjoint operator by subtracting from X its adjoint and dividing by two. If Q(X) denotes this anti-Hermitian operator then
a(x)=x+½ divX,
(7)
where the divergence of X can be expressed in terms of the Riemannian connection V on M as div X=e~(VxaX) , with {e a} the coframe dual to {Xa}. The commutator algebra of these anti-Hermitian operators is isomorphic to that of the Lie algebra of vector fields. That is
Volume 156, number 1,2
PHYSICS LETTERSA (8)
[Q(X), Q(Y) ] = Q ( [x, Y] ) .
For a coordinate vector we can alternatively write Q(O/dx a) as
0
Q
,0
log tr,
= Ox----~ + ~ Ox---z
with tr as in (3). To simplify the notation we shall write
Qa = Q(O/Oxa) • The classical Hamiltonian for a free particle moving on a Riemannian manifold is
H----½gabpaPb ,
(9)
where gab are the components of the metric tensor in some coordinate basis, with {p~} the conjugate momenta. The Schr6dinger representation of the canonical commutation relations in arbitrary coordinates is
xa-'x ", pa--,-ihQa.
(10)
Clearly if we transcribe the variables in the classical Hamiltonian as above then the resulting operator will depend upon the classical ordering. Furthermore such an operator will depend upon the coordinates used in the transcription. I f we order the classical Hamiltonian as H = ½Pagabpbthen the SchriSdinger substitution will result in a Hermitian operator. We get
QagabQb -- - h 2 l k + VQ,
( 1 1)
where A is the Laplacian on functions and the "quantum potential" function V0 is expressed in terms of the connection coefficient as [ 7 ]
4 gQ = -- h 2( 2 O~-~Faba-- 2Fa abFcbC--Fa OaFcbC) . (12) Indices are raised and lowered with the components of the metric tensor. The form of VQ makes explicit the dependence of this operator upon the coordinates used. The other ordering of the operators that, upon transcription, gives a Hermitian Hamiltonian is gabp~po+pbpcg'b. This is related to the previous ordering by
3 June 1991
gabQaOb+ObQagab =2OagabQn-h2(~-~bFa~+ o~bFaab) .
(13)
Both of the Hermitian operators that result from the classical Hamiltonian with the Schr&linger substitution depart from an invariant operator, the Laplacian, by terms involving the connection coefficients and their derivatives. A judicious combination of these differently ordered operators enables the derivatives of the connection coefficients to be combined into an invariant, namely the curvature scalar. We have
½QagabOb+ ~ (gabOaOb + abOag ab) = _ _ ~ 2 ( / ~ -- I R . ~_ I F a b C F c a b ) .
(14)
SO by choosing a particular ordering for the classical Hamiltonian we can obtain an invariant operator by augmenting the Schrtidinger substitution with the addition of a "quantum potential" that involves the connection coefficients but not their derivatives. This results in the "quantum Hamiltonian"
HQ=-h2( A - ~ R ) .
(15)
4. Conclusion
If we transcribe the classical Hamiltonian for a free particle using the SchriSdinger representation then the resulting operator depends upon the classical ordering. There is one, and only one, ordering such that the resulting operator is Hermitian and can be rendered invariant by the addition of a "quantum potential" 14h2FabCFcabthat involves the connection coefficients but not their derivatives. At any point such a potential will vanish in geodesic coordinates. The resulting quantum Hamiltonian departs from the Laplacian by one quarter of the curvature scalar. O f course the above prescription is only one possible motivation for adding a curvature term to the Laplacian. What is interesting is that it does single out one particular curvature term, and that this is not the ~R often encountered. One should, perhaps, also be guided in the choice of quantum operator by the requirement that the
Volume 156, number 1,2
PHYSICS LETTERSA
3 June 1991
c o m m u t a t o r algebra of some set of operators be isomorphic to a Poisson bracket algebra. Thus as well as the canonical c o m m u t a t i o n relations for the p's and q's we would also like
a BP Venture Research U n i t Fellowship. I.M. Benn is grateful to the D e p a r t m e n t of Physics, University of Lancaster for its kind hospitality.
[Qa, HQ] = Q({p~, H})
References
,
for any choice of coordinates. O f course this requires that we have an expression for Q({Pa, H})! There is a natural way to regard symmetric tensors on M as functions on T*M. This establishes a correspondence between Poisson brackets of certain functions on T*M and a natural Lie algebra on the symmetric algebra of such tensors [8,9]. We believe that Q extends to a h o m o m o r p h i s m on these symmetric tensors in such a way that Q({pa, H}) and HQ in (15) arise naturally and will report on these aspects elsewhere.
Acknowledgement This work has been carried out with support from
[ 1] P.A.M. Dirac, The principlesof quantum mechanics (Oxford Univ. Press, Oxford, 1958). [2 ] W. Pauli, Wellen-Mechanik,Handbuch der Physik, Band 23 I (1933)p. 120 [transl.: General principles of quantum mechanics (Springer, Berlin, 1980) p. 41 ]. [3] Y. Zhan, Phys. Lett. A 128 (1988) 451. [4] S. Dowker, J. Phys. A 5 (1972) 931; Ann. Phys. (NY) 62 (1971) 361. [5 ] S.I. Ben-Abrahamand A. Lonke, J. Math. Phys. 14 (1973) 1935. [6] H. Stephani, General relativity (Cambridge Univ. Press, Cambridge, 1985)p. 53. [7] B.S. DeWitt, Phys. Rev. 85 (1952) 653. [8 ] F.J. Bloore and M. Assimatopoulus,Int. J. Theor. Phys. 18 (1979) 233. [ 9] A. Sudbury, Math. Proc. Cambridge Philos. Soc. 81 (1977) 133.
PHYSICS LETTERS A
3 June 1991
A note on the quantum Hamiltonian for a free particle I.M. B e n n Department of Mathematics, Universityof Newcastle, Newcastle, New South Wales,Australia
and R.W. Tucker School of Physics and Materials, Universityof Lancaster, LancasterLA1 4YB, UK
Received 22 March 1991; accepted for publication 8 April 199 I Communicated by J.P. Vigier
We reconsider the non-relativistic quantum mechanics of a free particle on a Riemannian manifold and offer an argument for an invariant quantum Hamiltonian proportional to the Laplacian amended by one quarter of the scalar curvature oftbe manifold. An explicit ordering of the arbitrary canonical variables in the classical Hamiltonian is found that implies the existence of a quantum potential that can be made to vanish at any point by adopting geodesic coordinates for that region of the manifold.
1. Introduction It is a sobering thought that there is no clear consensus as to the correct quantum description of a nonrelativistic free particle moving on a Riemannian manifold. Even for a particle in R 3 there is no known unambiguous transcription from the classical to the q u a n t u m Hamiltonian. The Schr6dinger representation o f the canonical c o m m u t a t i o n relations leads to operator ordering ambiguities, and the q u a n t u m Hamiltonian obtained depends u p o n the coordinates in which the transcription takes place. However, in this R 3 case it is generally agreed that the transcription should result in the "Laplacian" on normalisable state functions as Hamiltonian operator, and ad hoe prescriptions can be given to ensure that this is the case when quantising in other than Cartesian coordinates [ 1-3 ]. The above ambiguities are far more troublesome for the canonical quantisation o f a free classical particle moving on a geodesic o f an arbitrary Riemannian manifold. In this case we cannot simply introduce extra prescriptions to ensure that the "correct" q u a n t u m Hamiltonian is obtained in that it is far less clear what the "correct" Hamiltonian is. Specifically
one can add curvature scalar terms to the Laplacian. Indeed it has been widely argued that the appropriate Hamiltonian operator differs from the Laplacian by the addition of~R, where R is the curvature scalar [4,5]. Ultimately the arguments for this curvature term hinge on obtaining the "expected" wave equation in "special" coordinates. These special coordinates {ya} are chosen such that at their origin the connection coefficients Fabc vanish, with the derivatives satisfying O F~/+~O F '% O -Oy -~ Oy ~ ca b-fTy~Fab ~= 0 . Although such coordinates are often referred to as normal we prefer to reserve this terminology for geodesic coordinates. These special coordinates are not geodesic coordinates, although they are closely related to them [6 ]. If the canonical commutation relations are realised through the Schr'6dinger representation then the ambiguities inherent in the transcription from classical to q u a n t u m Hamiltonian can be resolved by requiting that we not only transcribe the classical coordinates and m o m e n t a to their q u a n t u m counterparts, but in addition add a " q u a n t u m po-
0375-9601/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)
1
V o l u m e 156, n u m b e r 1,2
PHYSICS LETTERS A
tential" to the classical Hamiltonian [ 7 ]. This quantum potential will depend upon the coordinates in which the transcription takes place as well as upon the ordering of the variables in the classical Hamiltonian. Of course the ambiguity now becomes the choice of quantum potential. For example, we can essentially declare the quantum Hamiltonian operator to be the Laplacian and then choose the quantum potential to be the difference between the Laplacian and what is obtained by replacing classical p's and q's with their quantum counterparts. The quantum potential will then involve the connection coefficients, for the coordinate basis in which the transcription takes place, and their derivatives. We shall show that there is a unique ordering of the classical variables so that the SchrSdinger substitution for the p's and q's may be followed by the addition of a quantum potential involving the connection coefficients but not their derivatives to result in an invariant Hermitian quantum Hamiltonian. This Hamiltonian differs from the Laplacian by a ~R term.
2. Wave functions and densities
A direct comparison of papers on this subject is inconvenienced by the fact that some authors represent the state of the particle by a complex function on the configuration manifold, independent of any coordinate system, whereas others represent it by a complex half-density which, upon a coordinate transformation, changes by the square root of the determinant of the transformation matrix. For the case of an orientable Riemannian manifold these two approaches are equivalent as, to facilitate comparisons, we now show. Let M be an n-dimensional orientable Riemannian manifold with metric tensor g and orienting volume form • 1. Then a Hermitian inner product on complex functions ~ and ~u is defined by (¢, ~,) = f 0 " ~ , 1 .
(1)
M
The volume n-form • 1 can be written in coordinates {M} as • 1 =o'dx
where
I ,
(2)
3 J u n e 1991
a= Ideggol ~/2
(3)
and d x I = d x 1 AOx2A ...A d x n. If O=a'/2(~ then ~ is a half-density. A Hermitian inner product may be defined on half-densities by (~, ~ ) = j ~q)dx ~
(4)
M
so that
(¢, ~) = (~, ~ ) .
(5)
If 0 is any operator taking functions to functions then an operator taking half-densities to half-densities is given by
O=al/20a-l/2.
(6)
This relation between operators gives an isomorphism of the algebra of operators in that ( O ( 02 ) = 0102. It also readily follows that 0 is selfadjoint with respect to ( , ) if and only if 0 is selfadjoint with respect to ( , ). Thus, for this case of an orientable Riemannian manifold, any quantum description involving halfdensities has a parallel description in terms of functions. In the following we shall use the latter description and assume M compact to facilitate the definition of Hermitian operators.
3. The quantum mechanics of a free particle
An arbitrary vector field X is prevented from being anti-self-adjoint with respect to the inner product ( 1 ) due to a term involving the Lie derivative of the volume form. We can obtain an anti-self-adjoint operator by subtracting from X its adjoint and dividing by two. If Q(X) denotes this anti-Hermitian operator then
a(x)=x+½ divX,
(7)
where the divergence of X can be expressed in terms of the Riemannian connection V on M as div X=e~(VxaX) , with {e a} the coframe dual to {Xa}. The commutator algebra of these anti-Hermitian operators is isomorphic to that of the Lie algebra of vector fields. That is
Volume 156, number 1,2
PHYSICS LETTERSA (8)
[Q(X), Q(Y) ] = Q ( [x, Y] ) .
For a coordinate vector we can alternatively write Q(O/dx a) as
0
Q
,0
log tr,
= Ox----~ + ~ Ox---z
with tr as in (3). To simplify the notation we shall write
Qa = Q(O/Oxa) • The classical Hamiltonian for a free particle moving on a Riemannian manifold is
H----½gabpaPb ,
(9)
where gab are the components of the metric tensor in some coordinate basis, with {p~} the conjugate momenta. The Schr6dinger representation of the canonical commutation relations in arbitrary coordinates is
xa-'x ", pa--,-ihQa.
(10)
Clearly if we transcribe the variables in the classical Hamiltonian as above then the resulting operator will depend upon the classical ordering. Furthermore such an operator will depend upon the coordinates used in the transcription. I f we order the classical Hamiltonian as H = ½Pagabpbthen the SchriSdinger substitution will result in a Hermitian operator. We get
QagabQb -- - h 2 l k + VQ,
( 1 1)
where A is the Laplacian on functions and the "quantum potential" function V0 is expressed in terms of the connection coefficient as [ 7 ]
4 gQ = -- h 2( 2 O~-~Faba-- 2Fa abFcbC--Fa OaFcbC) . (12) Indices are raised and lowered with the components of the metric tensor. The form of VQ makes explicit the dependence of this operator upon the coordinates used. The other ordering of the operators that, upon transcription, gives a Hermitian Hamiltonian is gabp~po+pbpcg'b. This is related to the previous ordering by
3 June 1991
gabQaOb+ObQagab =2OagabQn-h2(~-~bFa~+ o~bFaab) .
(13)
Both of the Hermitian operators that result from the classical Hamiltonian with the Schr&linger substitution depart from an invariant operator, the Laplacian, by terms involving the connection coefficients and their derivatives. A judicious combination of these differently ordered operators enables the derivatives of the connection coefficients to be combined into an invariant, namely the curvature scalar. We have
½QagabOb+ ~ (gabOaOb + abOag ab) = _ _ ~ 2 ( / ~ -- I R . ~_ I F a b C F c a b ) .
(14)
SO by choosing a particular ordering for the classical Hamiltonian we can obtain an invariant operator by augmenting the Schrtidinger substitution with the addition of a "quantum potential" that involves the connection coefficients but not their derivatives. This results in the "quantum Hamiltonian"
HQ=-h2( A - ~ R ) .
(15)
4. Conclusion
If we transcribe the classical Hamiltonian for a free particle using the SchriSdinger representation then the resulting operator depends upon the classical ordering. There is one, and only one, ordering such that the resulting operator is Hermitian and can be rendered invariant by the addition of a "quantum potential" 14h2FabCFcabthat involves the connection coefficients but not their derivatives. At any point such a potential will vanish in geodesic coordinates. The resulting quantum Hamiltonian departs from the Laplacian by one quarter of the curvature scalar. O f course the above prescription is only one possible motivation for adding a curvature term to the Laplacian. What is interesting is that it does single out one particular curvature term, and that this is not the ~R often encountered. One should, perhaps, also be guided in the choice of quantum operator by the requirement that the
Volume 156, number 1,2
PHYSICS LETTERSA
3 June 1991
c o m m u t a t o r algebra of some set of operators be isomorphic to a Poisson bracket algebra. Thus as well as the canonical c o m m u t a t i o n relations for the p's and q's we would also like
a BP Venture Research U n i t Fellowship. I.M. Benn is grateful to the D e p a r t m e n t of Physics, University of Lancaster for its kind hospitality.
[Qa, HQ] = Q({p~, H})
References
,
for any choice of coordinates. O f course this requires that we have an expression for Q({Pa, H})! There is a natural way to regard symmetric tensors on M as functions on T*M. This establishes a correspondence between Poisson brackets of certain functions on T*M and a natural Lie algebra on the symmetric algebra of such tensors [8,9]. We believe that Q extends to a h o m o m o r p h i s m on these symmetric tensors in such a way that Q({pa, H}) and HQ in (15) arise naturally and will report on these aspects elsewhere.
Acknowledgement This work has been carried out with support from
[ 1] P.A.M. Dirac, The principlesof quantum mechanics (Oxford Univ. Press, Oxford, 1958). [2 ] W. Pauli, Wellen-Mechanik,Handbuch der Physik, Band 23 I (1933)p. 120 [transl.: General principles of quantum mechanics (Springer, Berlin, 1980) p. 41 ]. [3] Y. Zhan, Phys. Lett. A 128 (1988) 451. [4] S. Dowker, J. Phys. A 5 (1972) 931; Ann. Phys. (NY) 62 (1971) 361. [5 ] S.I. Ben-Abrahamand A. Lonke, J. Math. Phys. 14 (1973) 1935. [6] H. Stephani, General relativity (Cambridge Univ. Press, Cambridge, 1985)p. 53. [7] B.S. DeWitt, Phys. Rev. 85 (1952) 653. [8 ] F.J. Bloore and M. Assimatopoulus,Int. J. Theor. Phys. 18 (1979) 233. [ 9] A. Sudbury, Math. Proc. Cambridge Philos. Soc. 81 (1977) 133.