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INTEGRAL, Vol. 9 No. 2, Juli 2004

RELATION BETWEEN CLASSICAL AND QUANTUM

PHYSICS USING CANONICAL TRANSFORMATIONSSylvia H. Sutanto and Paulus C. Tjiang

Jurusan Fisika, Fakultas Matematika dan Ilmu Pengetahuan AlamUniversitas Katolik Parahyangan

Bandung, IndonesiaE-mail : [email protected]

Abstract

We discuss the relation between classical and quantum physics using the

algebraic structure of canonical transformation. The simplest relationbetween canonical transformation algebra and compact Lie algebra will beintroduced that leads us to a better understanding of the first quantization inquantum mechanics.

Keywords : canonical transformation, canonical realization, Lie group, Liealgebra.

Intisari

Dibahas hubungan antara fisika klasik dan kuantum dengan menggunakanstruktur aljabar transformasi kanonik. Akan diperkenalkan hubungan paling

sederhana antara aljabar transformasi kanonik dan aljabar Lie kompak yangmembawa pada pengertian yang lebih baik tentang kuantisasi pertama dalammekanika kuantum.

Kata kunci : transformasi kanonik, realisasi kanonik, grup Lie, aljabar Lie

Received : June 9, 2004Accepted for publication : June 21, 2004

IntroductionSymmetry of a physical system under agiven transformation will providedynamical information of the system suchas conserved quantities. The informationcontained in every classical system may beobtained by formulating the symmetry

properties under the canonicaltransformations. For every quantumsystem, the dynamical information of thesystem can be understood using the Lie

brackets (commutators) of operators whichform a Lie algebra. A certain class of Liealgebras, namely the compact Lie algebras

such as SO(n) and SU (n) algebras, playmany important roles in quantummechanics and quantum field theory.

In this paper we shall discuss theconnection between classical and quantum

physics using the canonicaltransformation. We shall firstly discuss thecanonical formalisms in Section 2. InSection 3, we briefly discuss that the set ofthe infinitesimal canonical transformationsforms an abelian group and algebra. Usingthe relation between Poisson brackets andthe Lie bracket, we shall derive the

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{ } M H M

q H p

M p

H q

M q

q M p

p

dM dt

,

=

= +

=

(2-12)

It is clear that if M (q, p) represents aconserved quantity , then. :

{ }0 , 0dM M H dt

= = (2-13)

Also, Eq. (2-4) can be rewritten as follows:

{ } { q q H p p H = =, ; , } (2-14)

If we decompose q anddF dt

of

Eq. (2-10) in terms of and and usethe integrability conditions of F (q, p ,t ), wewill have the following result [1,2] :

q p

{ } 1, =

= q p

pq

p p

qq

pq

(2-15)

Eq. (2-15) is the necessary and sufficientcondition for the transformation

( ) (,q p q p ), to be a canonicaltransformation.

3 Algebraic Structure of

Canonical TransformationsAlthough the full canonical transformationis not an abelian group in general, but aclass of certain transformations called theinfinitesimal canonical transformation G ={G }, where

{ }{ }

'

'

: ,

,

k k k i k i

k k k i k

G q q q q f

p p p p f

= + = + i

(3-1)

forms an abelian group under thefollowing operation [3] :

{ }( )

' : ,

k k k k i i j j

k

G G q q q q f f

G G q

= + +

= (3-2)

The function f i = f i (q, p) is called thegenerator of canonical transformation.The repeated indices in Eqs. (3-1) and (3-2) mean the summation over the indices,and i are infinitesimal parameters. Thegroup G also forms a linear vector spaceunder group operation defined in Eq. (3-2)and the following scalar multiplication [3]:

{ }'

: , j j j i jkG q q q q k f = + i (3-3)

Introducing the following vectormultiplication :

{ } { } ( )

( )[ ][ ][ ][ ]{ }{ }

: ,

: ,

:

, ,

G q q q q f

G q q q q f

G G q q G G G G q

G G G G q

q q f f

k k k i k i

k k k i k i

k k k

k

k i j k i j

'

'

''

= += +

=

=

= +

1 1

1 1

(3-3)

then the vector space G may form analgebra if and only if the followingcondition of generators is satisfied :

{ } ijk k ij ji d f K f f +=, (3-4)

where and d are constants, and the

higher order terms of i and K ij

k ij

in Eq. (3-3) have been neglected.

4 Canonical Realization of Lie Algebra

A group consisting elements withcontinous parameters is called a continousgroup, which is necessarily an infinitegroup. A continous group with

differentiable parameters is called a Liegroup. Since a Lie group is an infinite

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group, it is characterized by a finitenumber of elements X

k , k = 1,2, ... m,

called the generator of Lie group. A Liegroup wih parameters defined over aclosed interval is called a compact Lie

group. The set { X k } forms a Lie algebrawhose relation of generators is given by[4]

[ ], nk l k l l k kl n X X X X X X C X = = (4-1)

with are structure constants. Eq. (4-1)satisfies the following relation

C kln

, ,

, , ,

, , , , , ,

X X X X

X X X X X X X

X X X X X X X X X

k l l k

k l m k m l m

k l m l m k m k l

= + = +

+ +

0= (4-2)

with and are arbitrary numbers. Eq.(4-2) are also satisfied by the Poisson

brackets, i.e.

{ } { }{ } { } {

{ }{ } { }{ } { }{ } 0,,,,,,, , ,

,,

=+++=+

=

lk mk mlmlk

mlmk mlk

k llk

f f f f f f f f f

f f f f f f f

f f f f

}

) p

(4-3)

A canonical realization of Liealgebra is a relation between

( ,k k X f q (4-4)

which satisfies both Eqs. (3-4) and (4-1)[5,6]. From Eqs. (3-4) and (4-1), itfollows that the canonical realizations arenot one-to-one correspondences, becausethe constants K in Eq. (3-4) do not needto be the same as the structure constants

in Eq. (4-1), and also because of theexistence of the constant d . If we

consider the case where and thegenerator of canonical transformation f i transforms as :

ijk

C kln

ij

K C ijk

ijk =

i i i

k ij ij k

i f f f d

d C d

= +

= (4-4)

then we have

{ } k k ij ji f C f f =, (4-5)

where the relation between X k and f k isone-to-one correspondence. Such relationis called the true canonical realization .

Not every Lie algebra has an associatedtrue canonical realization because it is notalways possible to eliminate thecontribution of constant d in Eq. (3-4)through the transformation (4-4) in a givenLie algebra [4,5]. The compact Liealgebras, i.e an algebra constructed fromits associated compact Lie group such asSO(n) [6] and SU (n), may have their

associated true canonical realization.

ij

To obtain an explicit relation of Eq. (4-4),one has to solve simultaneously the set of

partial differential equations of the Poisson bracket relation (4-5) associated with Liealgebra with Lie structure (4-1). Thesimplest solution of (4-5) in accordance toEq. (4-1) is

( ) ( ),

,k m k mnm n

n f q p q X p= (4-6)

where ( )k mn X is the matrix representationof X k . In the form of differential operator,

X k may be written as

( ),

k m k mn

m n n

X q X q=

(4-7)

From Eqs. (4-1) and (4-7), it is clear that

( ) ( ) ( ) ( ) ( )n

k l l k kl nmr rn mr rn mnr X X X X C X =

(4-8)

As the example, consider the SO(3)algebra generated by the followingrepresentation of generators :

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1 2

3

0 0 0 0 0 10 0 1 ; 0 0 00 1 0 1 0 0

0 1 01 0 0

0 0 0

L L

L

= = =

(4-9)

satisfying the relation

3

1

,i j ijk k k

L L =

= L , (4-10)

where i,j,k = 1,2,3 and ijk is the three-object permutation symbol. Thedifferential representation of (4-9) using(4-7) is

1 2 33 2

2 3 11 3

3 1 22 1

L q q p p

L q q p p

L q q p p

= =

=

(4-11)

which coincide with the angularmomentum of 1-, 2- and 3-direction inthree dimensional quantum mechanics .

With Eq. (4-6), the associated canonicalrealization of SO(3) algebra is

( )( )( )

1 2 3

2 3 1

3 1 2

,

,

,

3 2

1 3

2 1

f q p q p q p

f q p q p q p

f q p q p q p

= = =

(4-12)

which may be interpreted as the angularmomentum of 1-, 2- and 3-direction inthree dimensional classical mechanics .The generators (4-12) satisfy the relation(4-5) with the same structure constant as inEq. (4-10).

Comparing Eqs. (4-11) and (4-12), or ingeneral, Eqs. (4-6) and (4-7), we maycome to the conclusion that

i i

ii

q q

pq

(4-13)

which is the famous first quantization ofquantum mechanics in configurationrepresentation.

5 ConclusionThe usual treatment of obtaining relation(4-13) in quantum mechanics is thecalculation of expectation value ofcanonical variables in configuration space.However, since the canonical realizationapproach gives the same relation, we maycome to the following conclusions :

The relation (4-13) may be regarded asthe condition to preserve the algebraicstructures in classical and quantummechanics.

The relation (4-13) allows us toreplace the Poisson bracket relation inclassical mechanics directly with thecommutation relation in quantummechanics.

The relation (4-13), together with Eq.(4-6), provides us information ofrelating any physical observable inclassical mechanics to the associatedobservables in quantum mechanics.For example, for any classicalobservables which is expressed in

( ), f q p , one may straightforwardlywrite the associated quantum operator

, f qq

.

The success of the canonicalrealization explaining the first quantizationencourages us to do the same treatment

between classical and quantum fields.However, since we have to deal withinfinite number of degrees of freedom infield theory, it seems difficult to do so. Ausual treatment in quantum field theory isto subject the commutation relation ofraising and lowering operators ofnonrelativistic harmonic oscillator to thequantum state functions (the so calledsecond quantization ) which is proven to bethe correct treatment in understandingquantum field theory.

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6. References1. Goldstein, Herbert, Classical

Mechanics , Addison Wesley,Massachusetts, 1980.

2. Sudarshan, E.C.G., N. Mukunda,Classical Dynamics : A ModernPerspective , John Wiley and Sons,

New York, 1974.3. Sutanto, Sylvia H., Struktur Aljabar

dalam Formalisma Kanonik , Thesis

S-1, Jurusan Fisika - FMIPA, InstitutTeknologi Bandung, 1991.

4. Gilmore, Robert, Lie Groups, Lie Algebras and Some of Their Applications , John Wiley and Sons, New York, 1974.

5. Pauri, M., G.M. Prosperi, J. Math.Phys. , 366, 1966.

6. Pauri, M., G.M. Prosperi, J. Math.Phys. , 2256, 1967.

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