Group theoretical methods in classical mechanics and continuum physics

Group theoretical methods in classical mechanics and

continuum physics

Citation for published version (APA):

Mooren, L. C. J. M. (1979). Group theoretical methods in classical mechanics and continuum physics. Technische Hogeschool Eindhoven. https://doi.org/10.6100/IR63280

DOI:

10.6100/IR63280

Document status and date: Published: 01/01/1979

Document Version:

Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)

Please check the document version of this publication:

• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.

• The final author version and the galley proof are versions of the publication after peer review.

• The final published version features the final layout of the paper including the volume, issue and page numbers.

Link to publication

General rights

Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain

• You may freely distribute the URL identifying the publication in the public portal.

If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement:

www.tue.nl/taverne

Take down policy

If you believe that this document breaches copyright please contact us at:

openaccess@tue.nl

GROUP THEORETICAL METHOOS IN

CLASSICAL MECHANICS AND CONTINUUM PHYSICS

GROUP THEORETICALMETHODS IN

CLASSICAL MECHANICS AND CONTINUUM PHYSICS

...

"

TrIl: - Rek", c' n

rum

ing.

16-2-'~

,.-'

oren.

I

ec.

'::60

'4

dOss./

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL EINDHOVEN, OP GEZAG VAN DE RECTOR MAGNIFICUS, PROF. DR. P. VAN DER LEEDEN, VOOR EEN COMMISSIE AANGEWEZEN DOOR HET COLLEGE VAN DEKANEN IN HET OPENBAAR TE VERDEDIGEN OP

VRIJDAG 2 FEBRUARI 1979 TE 16.00 UUR

DOOR

LEONARDUS CAROLUS JOHANNES MARIA MOOR EN

GEBOREN TE 's-HERTOGENBOSCH

Dit proefschrift is goedgekeurd door de promotoren:

prof.dr. L.J.F. Broer dr. B.J. Verhaar

Contente

Abstract

o

Part I

I. Discrete systems

1.1 Canonical transformations 1

1.2 Poissonbracket and invariant canonical transformations 4

1.3 The canonical group 5

1.4 Relation between finite and infinitesimal canonical

transformations 9

1.5 Structure of the algebra of tbe canonical group 14

1.6 Constants of the motion 27

2. Continuous systems 32

2.1 Canonical transformations 32

2.2 Invariant canon ie al transformations 36

2.3 Special transformations. Campbell-Baker-Hausdorff formulas

2.4 Relation between finite and infinitesimal canonical trans format i ons

2.5 Examples of canonical transformations and canonical transformations as realizations of finite-parameter group

Part 11

3. Classification of quadratic constants of tbe motion for

37

41

44

linear systems 60

3.1 The equation qtt - -V_q• General Theory 60

3.2 Discrete examples 75

3.3 Examples of continuous systems, scalar fields 81

3.4 Pseudo-rotations, pseudo-translations 86

3.5 Constants of the motion with a density explicitly

depending ont t 10~

Appendix A Appendix B Appendix C Appendix D Appendix E Appendix F References Samenvatting Nawoord Levensloop 114 120 122 125 129 131 137 139 140 141

Abstract

This work consists of two parts.

In part I canonical transformations are treated for discrete- (finite number of degrees of freedom) as well as continuous systems. Relations between finite and infinitesimal canonical transformations are

explored. For discrete systems the algebra of the canonical group is constructed out of the algebra of several subgroups. A whole family of Fourier- and Laplace-like transformations can be constructed which are realizations of some classical finite parameter groups. The proce-dure followed in this thesis is somewhat different from,the "usual" one: first the realization of the algebra ofsome group is found, then from this a realization of that group is constructed.

Part I I of this work deals with the classification of quadratic

constants of the motion for linear systems. The relation between commu tators of the dynamical operator and constants of the motion is inves-tigated.

It turns out that a class of equations which does not possess rotation-symmetry nevertheless has a rotation-symmetryoperator with rotation-like pro-perties. We call this a pseudo-rotation operator. Together with this pseudo-rotation operator there can be found pseudo-translation opera-tors forming a Lie-algebra which is isomorphic with the conventional algebra consisting of rotation and translation operators (ISO(2) and ISO(3}l.

Eigenfunctions of the pseudo-rotation operator are constructed in Fourierlanguage. Unfortunately all attempts to find the eigenfunctions

PART I

I. Discrete systems

1.1 Canonical transformations

In chapter I we shall treat Hamiltonian systems. The Hamiltonian will be a function of coördinate- and momentum variables (q(t) aod

pet»~ and of t. The functions q(t), pet) are, in general, vector-functions, although we shall not write the indices in most cases. The Hamiltonian H (p,q,t) is a scalar function.

The Hamilton equations are equivalent withthe equationsof motion

of the system, they are

( 1,0

qt means :t q(t). If no confusion can arise we shall not write down the explicit t-dependence of the functions. The functions q and p are cal led a pair of canonically conjugated variables. Let there be a

transformation from q,p to the functions q,p;q = q(q,p,t),

P

=

p(q,p,t)

We call such a trans format ion a canonical one if there exists a

function H(q,p,t) such that the equations we get from

H

0,2)

transformation q,p + q,p in (l,l) we get (1.2}. !here are two ways to construct a canonical transformation.

~) The first method is to use a generating function, a function of

one old and one new variabie (or in the n dimensional case: n old and n new variables). !here are four possibilities

We get the transformation in the usual way.

CJFl CJFl CJFl Fl P = oq , p

... - ar'

H • H

+ ät

CJf2 oF:z oF:z Fz p.~ _{. q} • q _{.. dp'}

H .. H +ät

oFa oFa oFa Fa q

_{- Ft> •}

_{p = -}

mr'

_H"'H+m:

aF .. oF ... Clf .. F .. q ..

_{- ä'P '}

q"~ ~

H ..

H + "li"i:

A canonical transformation of a special kind is the one wbich

transforms the Hamilton equations into.

tlH

₀ - = _qt

dP

aH

0 - = _{- P} t dq

!his is the case if

H

is a function of t alone. Generating this

transformation by Fl(q,q,tl we get:

aFl aFl ClFl

ii

=

H + "li"i: ' P = oq , p

= -

~

(J ,3)

(I,4)

(1,5)

or

0.5)

\be Hamilton-Jacobi equation. Let H be a given functiOn, a possible

soluti~ F1 of tha H.J.-equation tben is tbeaction integral

F1

=

J

(p qt - H)dt ([2])

o

*) tbe second method of construction is to start with an infinitesimal

canonical transformation (canonicalvariation). The identical

trans-formation is generated by F2 (p,q. t)::pq. A canonical variation can

then be generated by pq + 8g(P.q,t), g is called the generator of the

transformation (the variation as well as the fini te transformation ar~

constructed from tbis). The equation (1.3) gives

q

and

p

8 a small parameter, so we may neglect second order terms in 8. That means we may change g(p,q.t) into g(P.q.t). So we get:

every function g(p,q,t} is the generator of a canonical trans forma-tion and tbe first order variaforma-tion in q and p is

A tinite transformation ean now be eonstrueted by performing sueees-sive infinitesimal transformations. We get in this way a whole family of transformations parametrised by a parameter s by integrating the equations

dq(s)

=

~ dp(s) = _ ~ p(O)

=

P q(O)

ds op(sJ • ds aq(s) • , q (1,7)

g is a funetion of q(s) and p(s}. See seetion 1,3 for more details.

1.2 Poissonbraeket and invariant eanonieal transformations

Given two funetions f(q,p) and g(q,pl. The Poissonbraeket

L.J

between f and g is defined in the following way

L

f,gJ af •

2.K

_!! .

2.K

=

oq op ap aq (2.1)

ramark: qand pare vectors. so the terms on the right hand side of the equation are inner produets.

A special case

l

q •• p.

J

=

0 ~J" Let a eanonieal variation be generated

~ J *

oy

g(q,p,tI aeeording to (1,6J. we then define

oH,

the variation of

H.

oH.· H.

(p.q,tl -

H. (p,q,tl (2,2)

Up to firstorder in'~this ean ,be written as:

oH

aH

For

*"

we find an e.xpre.ssion, witl!.. q .and p evolved according to

0.11.

dg .;. '.2.& + d t a t

19,KJ

(2,n.

(2,4[. and (2,51 togat1ier: .~

-

-'-

.

, - - '

E.

dt = H:(p,q,t[ - H(p,q,tl - H(p,q.tl + H(p,q,t) -+

E.:~

-

H(p~Ci,tI

- Il(p,Ci,tI

Wa slia.ll caU tlie. trans format ion an invariant canonical trans forma-tion tf

i

is tlie. same. function of

p

and

q

as K of p and q, or:

ii(p,~;tL;' H(p,q;tL K(p,q;tI: == I:t(P.q,t)

We gat as a final re.sult: TIie. generator of an invariant canonical variation is a constant of t1ie. motien and vice versa.

1.3 The canonical group

Before going on we shall give fust a review of definitions of groups, . algeDras, representations and so on. We shall frequently use these

definitions and properties in the following sections (see for example

[3], [41).

A ~ is a set G of dements with one relation defined betwèen the

elements (relation • and with the properties:

a E G, bEG ~ a • b € G ; Va € G 3a-1 EG; 3 eEG with

-]

a • a = e. a • lol. "" e • a • a Va E G

A subgroup He G is a subset of G with a groupstructure itsel! and

the. same relation as G.

An

algebra A is a set of elements together with two relations (+ and

x) between the elements. If a, b, c

€

A then a + b E A and a x b E A,

further ax(b+c) • axb + axc.

An

algebra is an associative algebra if (axb)xc = ax(bxc). and a

Lie algebra ifaxa = Oand (axb)xc + (bxc)xa + (cxa)xb == 0 (the

Jacobi identity).

A subalgebra Be A is a subset of tha algebra Awith an algebra

struc-ture itself, a; strucstruc-ture with the same relations + and x. A

sub-algebra Be A is an invariant subalgebra ifaxb E B, Va E A and Yb € B.

Let A be an algebra and B a subalgebra of A: Two elements in A~ (A

mod B) are identified if their di'fference lies in B. This defines an e-quivalence-relation. The space A'\B is the space of equivslencè-classes. If B is invariant A\B is subalgebra of A. Tbe number çf linear inde-pendent elements of an algebra is cal led the dimension of the algebra. A representation R of a group G is a set of matrices R{a) with the

property R(a).R(b)

=

R(c) if a·b == c(a,b,c €

G).

In an analogeous way

the representation of an algebra is defined, the mapping preserves .the

algebrastructure (the relations + and x).

of operators, preserving the algebra strueture. We sha11 not a1ways distingish between a representation and a rea1ization if no eonfusion

iA possib1e (a representation is a rea1ization). If by a trans

forma-tion T all matrices R(a) € R(G) ean be brought in bleek form the

representation Rwi11 be eal1ed reducible, otberwise irreducible.

The funetions g(P.q) (We suppose, for eonvenienee, g not explicitly

~epending on t) eonstitute a Lie algebra with tbe relations + and "poissonbraeket". The algebra bas dimension infinity and we use for

this algebra the symbol Al' We eau ehoose a basis for Al by taking

the homogeneous functions of degree n: qlpk. n

=

k+l, n integer

~

O.

For convenienee we sball treat the ene' dimensional case, a

generali-zation to more variables (n-tuple q.) is straightforward.

l.

To every function g(p,q) we ean associate a linear operator g(p,q) in the following way

g(P.q)

=

ag

L

_!&

a

aq ap ap aq

g

operating on a function f(p,q} gives

i .

f(p,ql •

ig,fJ

special cases:

i .

q •

-* '

g •

p ..

~~

(3,1)

(3,21

remark: We llave restricted ourselves to functions g(p,q) wbich are analytie in a neighbourhood of tbe point (0,0).

This is sufficient for alocal theory. Even if g ean be eontinued analytically it is not sure that g generates global

transformations (transformations onto the whole phase space) see e.g. the example at the end of section 1.4.

The operators

g

constructed through (3,1) form a Lie algebra with

respect to + and the commutator [,J. This algebra Az is isomorphic

with[';;he algebra A111 (for isomorphism see e.g.)3], [4n~

Lg,fJ

=

h . .

[g,IJ -

n

A lil means that we must throw away the central element 1 from Al

(the element 1 has Poissonbracket zero with all g € Al, an element

with that property is called central). An additiona1 constant has no

influence on the re1ation.

From the algebra Aa we can construct a group of canonica1

transfor-mations in q and p. First we construct a one-parameter subgroup,

generated by

g.

q(t) = [exp (a

8) •

q] (tl

p(tl = [exp (a

g} •

pJ (t) (3,3)

This is a canonica1 transformation from q,p to q,p.,The transformations of tnis kind form a commutative (Abe1ian} one-parameter group. tf

T is the transformation (3.3i tnen: T • Ta • qaT 0 • q, T

O is

a a ~ M~

tne identity. !he exponent in formu1a (3.31 is a forma1 power series

exp a

g

00

I;' I n - n

L =r

a

g

n=O n.

All possibie products of these one-parameter groups (of different g) constitute a group of transformations. the canonical group, an infinite-parameter group.

As already mentioned problems can arise if (3,3)does~ot hold on the

whoie phase-space. if the transformation becomes singular for some

value of a (or if the mapping q.P ~

q,p

is not surjective).

In that case the one-parameter group under consideration is not a global Lie group. In a sufficientIy smail neighbourhood of the

identity (a "smaU enough") the transformation holds and we shall

speak of a Iocal Lie group. !his shows thet it is somewhat easier to consider the algebra of the canonical group rather than the group

itself. ComBining (3,3} and (3,4} gives:

1 2

[exp (a

8l •

q] (tl

=

q(tl+a h.qJ +

TI

a 19,1g,qJJ + ••• (3,S)

In the following sections we shail meet exampies of transformations which become singular for certain a-values.

Remark.:

!he generator of' a transformation T _{.... a} is given by

g

='~

T

I

as will Be clear from the foregoing.

a a a=O

1.4 Relation lie.tween· finite and infinitesimal canonical transformations •

The description of canonical transformations described in sections

1.1 and 1.3~st Be equivalent in some sense.

If g(q,p} is given, We get a finite canonical trans format ion through

(3,5). and we can ask the que.stion: what is the generating function F of this transformation?

Another question is the inverse: given a generating funetion, say Fz, what is g(q,p)?

!Wo points must be mentioned:

l} The infinitesimal approach gives transformations whieh are

continuously connected with the identity (Lie group). The trans

forma-tions generated by Ft(q,q} and F4(p,p) are not connected with the

identity, transformations generated by Fz(q,p) and Fa(p,q} have that

property and we shall therefore use Fz and Fa.

2} Fi (i

=

1,2,3,4) is parameter independent. We shall eonnect with

F. an one-parameter family Fr (-,',a} in sueh a way that F2(q,p) •

J. J.

F2'(q,p,aa) for a eertain value ao of a and F2(q,p,O) = qp, the

identity transformation (analogeous relations for Fa). Let F2(q,p} he

given, it is then clear thet we ean make the parametrization Fz'(q,p,a) in more than one way (in general} and the problem is not one-to-nne.

If. on the other band, g(q,p) is given we ean find the family Fz'(q,p,a) and the generating funetion Fz(q,p).

Let there be given a funetion g(q,p) whieh generates the transfor-mation

q(q,p,al = exp a g • q

p(q,p,al • exp ag- p (4,1)

Define a function F2(q,p,al (tba prime will be suppressed by now) such that

-

oF

a

p(q,p,a) .. oq

- - OF2

q(q,p,a) ,. dP (4,2)

are equivalent with (4,1) (af ter inversion).

We shall now derive an equation for the function Fa. We make use of

the fact that the Poissonbracket between two functions can be calcu-lated with respect to any arbitrary pair of canonically conjugate

variables. For every function f(q,p) the equation

:~

= -

Lg,fJ holds.

Especialy for q,p

(4,3)

g(q,p) is invariant under the transformation generated by itself, so

g(q,p) .. g(q,p) (from g(q,p) ,. exp a

g •

g(q,p) .. g(q,p».

On the other hand we have:

d / d é) /

a

I

F )dp

'I\"'" • ~ F2 + (,,- ;:;=r 2 d

op q,a oa op q,a op q,a a

Combining (4,3) and (4,4) we get

Cl / o F , .

~I

+ Clq/

~

1_ ..

og

1

'ijf q,a 3a 2 dP q dP q,a dq p dp q,a

In which g and q are regarded as functions of q,p and a,

g(q,p) .. g(q(q,p,a),p). In the same way we get

(4,.4)

and from this:

'2-1

aFz

=

~/_

aq p,a aa aq p,a

Cembining (4,5) and (4,6) gives the result

a

- I

.

Fz aa q,p

(4,6)

(4,7)

The funetion ~(a) is not essential, every g + ~(a) (~ arbitrary) gives

the same trajeetory in phase-spaee, so we ean put ~

=

°

without any

restrietion. We then arrive at the desired equation for Fz:

(4,S)

Equation (4,71 must oe aeeompanied by the initial eondition for a

°

-qp qp.

Ifa.=O, then

(4,8)readS~~Z

1

a=O g(q,p(q,p,01J=g(q,pl. It is possible

that for a eertain value of a the transformation generated by

Fz(q,p,a} oeeomes singular. In that case we ean use F3(p,q,a), beeause'

Pz and 1"3 are,not singular simulta'neously «(2], see for instanee the

example at the end of this sectionl. When we take g = -H (The

Hamil-tonian~a.=t then(4,S[ is the Hamilton-Jaeobi-equation. We ean regard (4,Sr as a generalization of the H.J.-equation and so Fz as the

generalization of the action-integral. Tbis is the case because the

equations (4,3) are analogeous to the Hamilton equations, with a

instead oft and g instead of H. A canonical transformation from

begin -(tsO) to end-position (t) is generated by the action integral, a weIl chosen solution of the H.-J.-equation. With this in mind we

find an expression for Fz

a

F2

(q,p)

=

J

(q

~

+

g(q,p»

da + qp

o

(4,9)

Tbe constant term on the right hand sidè of (4,9) is necessary to fullfi11 the initial conditions. Relations similar to (4,8) can be

derived for each F_k,' E.g. the relation for Fl(q,q} reads

= -

(4,10)

Tbe boundary conditions must be chosen in a region where Fl is

non-singular, Dut this is not possi'Dle in tne neighbourhood of the

iden-tity. For tOa derivation of (4.31. (4,9) and (4,10) see also [5], [6].

Example: g(q,PL

=

pq2. !lie equation (4.81 now is

aF_z __ _dF2

aa

az

and from tli.is F2

=

f(z-al h(p) + b(p}. Now usa the initial

h = kp, b = 0 (k constant)

There is a singularity at a •

~.

for tbis value of a use F3

- - -1

F!(p.<l.a)= -pq(l+a.q)· whicb is certainly not singular at a ,. -

.!.

q

Formally (4,8) determines tbe solution Fz. It is. of course,

some-times very bard to solve tbe equation.

1.5 Structure of tbe algebra of tbe canonical group.

The canonical group is thej?;roup cotresPQ.ndinl!lwitb::the,alge~ra with basis

1 k

q p ,land k integer ~ O. We shall form subalgebras and construct

tbe endre infinite-dimensional algebra frOlU finite-iIimensional

suh-algeDras. For tne canonical group of dimension n we use the symbol •

K(Znl. Th.e algelira of a group IC is L(Kl.

We shall treat tne algeBra L(IC(ZII.

!he set qn, n integer ~ 0 is an aoelian suoalgetira of L(IC(ZU

<lqn,Q1mJ ,. 0) generating the following transformations:

óq ,. 0

n-l óp ,. e:: n q

forma-tion of this kind has the property that the fir~t variation in tbe

Lagrangian L is a total time derivative, oL =

~t'l!

(q,q) (provided L

.~

6

a

2_L

has the form q q - f(q) and

äqT

~ 0). Such a transformation leaves

the equations of motion invariant and L is "gauge-invarèant"

the transformation. The action Intp;'gral is invariant; 15

J

Ldt

under

'" 0.

h h n . 0 ' h I b O f

In t e same way t e set p , n ~teger ~ ~s t e a ge ra 0 the gauge

group in q, Y (an abelian group toa).

q

Another subgroup is generated by the al~ebra pqn, oq '" Eqn, op '"

n-I (' d" f ( ) ' 1

-E n pq ~n more ~mens~ons Pi q, L '" • • n, f a function

af of q and Qqi '" Ei f(q) , op. 0= -E.p. -,,-).

~ J J "qi .

This group is the group of coördinate transformations Qq (the vari-ation in q is a function of q only}. A realizvari-ation of tne algebra

L(Qq} is xna _x (or Xln1 ••• X. ~ d , i

=

1, ••• , k ; k-dimensional).

K X.

L

!he set qpn, n integer ~

°

forms an algebra. !he group corresponding

herewith is the group of momentum transformations Q ;

p

~ n-I" n!he 1" k-d' . .

uq = E n q p ,up '" -E P • genera LzatLon to LmenSLOnS LS

as before. We can check immediatelythat Qp and Qp are groups,

for lpqn, pqmj

=

(n-rnJ pqn+m-I.

an element of L(IC(2H. On the other hand: every element of L(K(2}} can

be written as a Poissonbracket la,bJ with a € L(Qql and b € L(~). We

can formulate this as follows:

We ean also very simply prove the following statement: L(K(2» is the algebrale closure of L(Y ) and L(Y ). Por the n-dimensional case

q p ,

the same is true, the proof is somewhat more complicated but still simple.

Finite dimensional subalgebras are

I) !q2, !p2. qp with relations liq2, !p2

J ...

-qp,

L!q2,pqJ ... q2, llp2.qpJ ... _pi, forming ~n algebra isomorphic

with the algebra of SU(2i or SL(2,R}. Thisset is the algebra of

the group SP(2}. the group of linear canonical transformations,

I

'0 I,

leaving the s;YlnPlectic matrix S'" -1'.0 invariant, this means

t

a S a ' " S. a € SP(2).

SU(2) is the group of unitary 2x2 matrices witb determinant equal

to one. SL(2,R) is the group of real 2x2 matrices with determinant

equal to 1.

2) p, pq, ~pq2, finite subalgebra of L(Q ):L(C ). _{q q}

e

_q stands for the group of conformal transformations in q. In this one-dimensional case: the translation, dilatation and the special conformal trans format ion (generated by !pq2). For more details see the end of this section.

3) q, pq. !qp2, fini te subalgebra of L(Qp): L(C_p)' Cp the group of

conformal transformations in p.

L(K(2)} is the clusure of L(C ) and L(e ), everyelement

q p

a € L(K(2» can be formed by (repeated) Poissonbrackets between

e1e-ments of L(e ) _q and L(e ). _p We shall write K(2) ...

e*

_q C for the groups _p corresponding to the algebras, K(2) is the "product" of tbe groups

C and C •

1~e ean prove the foregoing statement by induction:

llpq , !qpJ c q p , lqn-l pn, q pJ • _2qnpn+l etc.

n n+1 k 1

From every q p we ean construct an element of the form q p by

forming lqnpn+l. q(p)J, and this several times.

Every g(q,p) generates a one-parameter group of transformations. If we take g2, g3 or f(g) resp. we obtain the same trajeetory in phase-space as generated by g, the only difference is the velocity with whieh the trajeetory is passed. (It is possible that for a certain value of the parameter the trajeetory stops, the transformation beeomes singular.) We ean see this immedia tly •

2"

~a ~a

(g} 2g

g, g

=

dq

3P - apa,q

The set g,g2 ,g3, ••• is the generator set of an infinite parameter group of transformations in phase-spaee, all possessing the same

trajeetory. The group is abelian. we call this group the

trajeetory-f 1 ( ' ) -

a

a

group 0 g. Examp e: g q,p • qp, g .. p ap - q aq

exp a.

g

q

..

q·e-a. q(a.)

p p·ea p(a.)

exp a. g2

I:

q'e -2apq

p·e2a.pQ

J q q2 q3 q" _" "

..

" ..

p qp q2p q3 p p2 qp2 q2pZ p3 qp3

p"

We ean now representa trajectory-group by a straight line through the central element 1. Every element on this line is a generator of the trajectory-group.

(3)

(g)

(I)

(I) is the trajectory-group of qP. (21 is the t-group of pqz. '(31 that

of q, etc. Every set qkpl

with~

a fixed rational number is the set

generators of a trajectory-group.

Every element of the algebra L(K(2}} generates a mapping of the

alge-bra into itself. If we use Ig > for an element of L(K(2» the mapping

ean be defined in this way: h

!

g >

=

I lh,gJ> (or h.g

=

h.g). With respect: to tlie basi.s of the algebra consisting of the homogeneous

polynomial s the 1!lapping is a linear representation of the algebra

(infinite dimensional matrices). I f the mappingis anto h is an

automor-phism', moreprecj.sely an innerautomorphism: (an outer automorphism of the algebra is an automorphism due to operators not contained in the

alge-bra itself).

Remark: For groups too there is a "natural action" of the group into ieself: define a

I

b > =

I

a'b> , a. b E K(2). This leads

- to the known concept of adjoint representation for groups~ In

analogy we can speak of the adjoint representation of an alge-bra.

Examples of automorphisms of L(K(2) are

d + 0 0 0 0 0 f' h f '

-q oq p op. q op' p oq'

ä'P'

ä'q'

love operators, togee er ornung an

algebra isomorpbic witb L(ISP(2» (ISp(2) is ebe inhomogeneous

symplec-tic group in two dimensions, the symplecsymplec-tic transformations together

with translations in p and q).

Tbree dimensional case.

We shall construct L(K(6l}. an infinite'parameeer algebra, from a

num-ber of finite-dimensional subalgebras with known properties and

reali-zations. Besides the algebralc closure we use a few more concepts:

Tbe direct product of two groups A and B is a group C. such that

every element of C can bewritten uniquely as a product of an element of A and one of B.

C .. A cS) B = {c

I

c - a'b " a € A" b EB" a'b .. b ·a}

Tbe algebra L(C) is the direct sum of L(A} and L(B); L(e) ..

L(A}

e

L(B} means LL(Al. L(B>J

=

0 and LL(A).L(A)J ~ L(A),

LL(B}, L(B}J ~ L(B}. For semi-direct sum or product we use the

symbols

e ,

_s ® • _s C .. A <B> _s B means A is an invariant subgroup of C, or

-1

e a c E A,

va

E A and \Ic E C. Au element e € C ean not be written

.as a'b in a unique way.

Tbe corresponding relation for the algebras is: LeC) .. L(A)

e

L(B),

s '

We can construct subalgebras of L(K(6)) and so subgroups of K(6):

*) !he set Ti - Pi (i=I,2,3) translations in qi' generators of an

abelian three parameter group,

*)

{Ti}

=

{qi} translations in Pi' generators of a group isomorphic

with the three-dimensional translation group. subgroup of the Galilei-group.

*) D • Piqi' generator of the diiatation group. the group of scale

. -a a

transformat~ons qi + qie ,Pi + Pie ,

*) {R.} = {E"k q.P_k}, generators of the three-dimensional rotation

~ . ~J J

group 50(3), or more precisely: a canonical realization of this

group. !he defining, three-dimensional, irreducible representation of

this algebra is

I

' , '·1

' • 1

,,-1 • .~.

, ,-I 1 •

!he action on (q 1 ,q2 ,q3) is:

R· exp aRt' exp

B

_{R2, exp} y~, col(q),q2,Q3)

=

o

o

cos

e

0 -sin 6 cos ..

y

sin y 0

o

cos a sin ex 0

o

-sin y cos y 0

o

-sin ex cos a sin

B

0 cos 6 0

o

and the action on

p

is quite similar.

p

+

R'p.

We can choose another

but this is not :essential. The latter will be cal led a canonical

-parametrization. It is because R is an antihermitian operator that p

-transforms in,the same way as q.

*) {M •• }

~J {q.p.- ;. qlPlo .• }, just a's before the summation convention ~ J ~J

is used. M .. are the generators of the special linear group SL(3,R),

1.J

the group of rea I 3x3 matrices with determinant equal to one(without

the term qlPlOij we get the generator set of the group L(3,R), the

linear transformation)

*) {Ki} - {QkqkPi-2QiClkPk}' the generators of the special conformal

transformations in

q.

LK.~.J - 0, so the K. are the generators of an

~ J 1.

abelian group.

*) {Ki} - {PkPkqi-2PiqkPk}' the generator-set of the special

confor-mal transformations in

p.

Ti' D, Ri and Ki form an algebra, the algebra of the conformal group

in three dimensions (a ten parameter group). Thecommutation relations

are: LR. _1. ,R. _J

J

= 1;:. ' •• _{1.J ....} 1L _-1<. LR.

,DJ

=

LT.,DJ

=

LT.,T.J ~ 1. ,1. J LR.T.J .. e:"k T_ 1. J 1.J l t LK. ,DJ .. K. 1. 1. LK.,K.J '" 0 1. J

LKpll)

=

E:_{ijk ~} LIL;T.J .. -20 •• D -2e:"k R. ~ . J ~J ~J-1<.

The infinitesimal transformations in qi and Pi are:

T. : Cli ... qi - e: 0 ••

J ~J

Pi ... Pi

R. : _{qi ..,. qi - e: jik qk} J

D Pi -T Pi - E j ik Pk qi ->- qi - Eqi p. ->-1 p. 1 + EBi K. : qi ->- q. + E (2 qiqj - óij qkqk) J 1

Pi ->- Pi + E (2 qiPj - 2 qjPi - 2 óij qkPk)

Remark: The finite transformations generated byK. are: 1

,..". ... ,.,. ... ... "'-1

q. ->- exp (a.K.)·q. = (q. - a. q-q) (1-2a'q + a"a q"q)

1 JJ 1 1 1

The expression for exp (a,K.)·p. is fairly complicated, this J J 1

will be computed (just as the transformation of qi) in appendix A.

Quite similarlvTi, D, Ri' Ki are the generators of a group, isomorphic with the ten parameter group of conformal transformations in p, the

group Cp' The commutation relations 'are

LR. ,R.

J ..

E:" k R. 1 J 1.J-1<. LR. ,TlJ .. -E:. 'k Tk' 1. J 1.J LK!,DJ .. -Kl 1 1. LT!,T!J

=

LR.

,DJ 1. J 1. LD,T!J

=

T! 1. 1. LR. ,K!J - E:" k K-' 1. J lJ-1<. LK! ,T!J 2(ó .. D - E" k R.) 1. J 1.J l J - l t LK! .K!

J ..

0 1. J

!he infinitesimal transformations in q and pare:

Tl: q1.' ->- q1.'

J

Pi"" Pi + E Ó ij

Kl: ql' ->- ql' + E: (2p.q. - 2p.q. - 2ó··qkPk)

p. + p. + e (PkPko .• - 2p.p.)

1 1 1J J 1

D and R.: as before.

1

The finite transformations exp (a.·K!)·p. c.an be c.a1c.u1ated in the _{J --1 1}

same way as donein appendix A.

... ... ... -1

exp (a.'K!)·p . • (p. + a. P'p) (1 + 2a.p. + a-a P'P)

J J 1 1 1 J J

The group C bas two Casi:Jnir operators (homogeneous po1ynomia1s,

. q

. bui1d from tne generators and commuting with all of them). A general

expression for the nth order Casimir operator is:

with

I.

the generators of the group and

1

k

.Here the structur&<eotlstants c. of the group are defined by.

ij

The metric tensor of the group algebra is g with components gij' defined by

~ ~ i

ij

A sufficient condition for the existence of g is

11

g ••

11

:(0 0

~J

A group (algebra) is cal led simple if there exists no invariant proper

subgroup (subalgebra), and semi-simple if there is no invariant abelian proper subgroup (subalgebra).

lor simpie and semi simple groups Ilg .• 11 :(00 ([3]). The conformal

lJ

group has no invariant proper subgroups, so it is a simpie group and gij exists. The conformal group in three dimensions is 10caIIy

isomorphic with the group 80(4,1) (the group of orthogonal

trans-formations Ieaving the metric diag. (+1,+1,+1,+1,-1) invariant).

This ~roup has two independent Casimir opé~atQ~s (see e.g.

[3]). It tnrns out that tbe only non-identically zero operators are

C

₂ and

c

4•

C

2 can be simplified using the definition of gij

C

-

=

ij 2 g X.X. 1 J

We find for

C

₂ and

C

4 af ter some calculation:

C

2 =

i<T.i.

1 1 +

i.T.}- R.R.

1 1 1 1 +

15 15

These operators are zero in this special realization. The two casimir

operators for Cp are

C

₂

and

è

₄

with all generators replaced by their

primed analogues, tbis means that C

₂

and C

₄

are zero two in the

cano-nieal realization. The conformal group in four dimensions (x,y,z,t) is treated e.g. in [8].

The algebrale closure of L(C_q) and L(SL(3» gives the set

Pl,qiPj,qiqjPl' ••• , qi ••• qkPl •••• , or the set f(q) Pi with f a polynomial in qi' This set is the generator set of the group of eoördinate transformations (or the "genera 1 eovariance group"; in the four dimensional "t.elativistie" case this

nO n1

transformations with generators ~ XI

is the group of coordinate

n

2 n3

x2 x3 0x.' [7],[8]).

l.

The closure of L(e ) and L(SL(3)} gives in the same way the algebra

p

g(P)qi' g a polynomial in Pi' The corresponding group will be called the "general contravariance group". For the proof ofthe two statements above see appendix B. The closure of the algebras of the general covariance - and general contravariance group gives the algebra L(K(6», this is analogeous to the one-dimensional case and the proof is straightforward.

We can set up a diagram for the algebra L(K(6}) or the group K(6). The dimension of every group (numher of generators) is given at the lower left corner of the boxes. The convention is in a semi-direct product

to write down first the invariant subgroup, so ISO(nl

=

Te

SO(n). ;

s

(ISO(n) is the inhOmb"geneous ó~thogOnai group in n dimensions; T,the

n~dimensinnal'abelian translati6n:gro~p, SO(n)'the group of orthogonal

fransfórmat:ions}~l"Further;.A"" B !!' C means L(er is the closure ,of

L(A) and L(B).

Canonical group K(2n) K(2n) = COV * CONTR.

00

'COV: general covariance group C

*

SL(n) q 00 C :conformal group of q Cq=K*[T® (SO(n)~)] q s Hn+l) (n+2)

CONTR.:general eontra~arianee group

C _p

*

SL(n)

linear C confo~l group of p.

. P=K*[T'® (SO(n)®D)]

p s

!~n+l)(n+2)

K:special conformall T®s(SO(n)®D) transformations 1 , - - - " , . K ":special 'éonformal transformations of p n Hn+l)n+l T: n ln(n-l)+l n

We now return to K(6). There are several other subgroups of K(6) with

physical importance. E.g. take the set H = !p.p.,R. = E .• kq.Pk'

L L L LJ J

P'i =Pi' Gi =qi' I= 1.

The commutationrelations are

lH,P.J = lH,R.J = 0

L L

lG.,p.J = 0 •• 1.

L J LJ

LH,G.J = -Po _{L L}

The algebra formed by this set is isomorphic with the algebra of the

extended Galilei-group ([9],[10]), that is the Galilei-group extended

with the central element 1 (LG.,p.J = 0 .. I in stead of LG.,p.J = 0).

The interpretation is simple: H is the Hamiltonian of a free. nonrela-tivistic particle without spin, R. are the generators of the rotation

~

subgroup (orbital angular momentum) and P. are the components of the

. J

linear momentum of tbe partiele; G_i at last are the generators of the

pure Galilei-transformations.

1.6 Constants of the motion.

From two constants of tha motien we can build new ones by taking the Poissonbracket or the product. The set of cORstants of the motion forms

a Lie algebra with respect to

L,J

and also an associative algebra

with respect to x.

Given a Lie algebra A, by forming products of elements of A we get an

algebra Ä, an algebra with the relations

l.

j,

x (and +). We shall

call

Ä

(as usual) the universal enveloping algebra of A ([11]).

Although tbe elassificatien of the constants of the motion will be treated in Part II, we shall give here a few simple examples: the two dimensional harmonic oscillator, isotropic and anisotropic ([12]). The three dimensional isotropic oscillator. is treated in [13).

*)

The two dimensional harmonie oscillator, isotropie, Mass and

frequenty are choosen equal to one for convenience.

There are four linear independent quadratic constants of the motion: Hand further

A

=

i(pt + q~) - i(p~ + q~)

M = P2ql - Plq2 P

=

PIP2 + qlq2

The four operators

H, A,

H,

P

working on eol(ql,pl,q2,p2) have a four dimensional matrix representation (the defining representation of

Sp (4);

H, A,

Mand Pare the generators of a subgroup of Sp (4)) •

0 0 !6 -1 0 -1 0 0 0 !6 !6 -} 0 I· !6 !6 .,.} -1 0 0

o

-1

o

o

o

o

o

-} -1

o

H

generates rotations in ql,pl and q2,P2 plane simultaneously, À

generates rotations in the same planes, eloekwise and eountereloek-wise respeetively,H generates rotations in Ql,Q2 and Pl,P2 plane and

P at last rotations in QhP2 and Q2,Pl plane simultaneously. A,M,P are generators of a group isomorphie with 8U(2). The four dimensional representation of 8U(2) dedueed from the representation above is redueible and ean be redueed to two irredueible representations of

8U(2) (or U(2) if H is ineluded). The symmetry group of H is U(2).

Remark: The four dimensional representation of Sp(4) is irredueible. The algebra of eonstants of the motion has a basis AnI Mn2 pns, ~I,n2, ns integer,:: O. The transformations generated are regular in the whole phase spaee.

H2

=

A2 + M2 + p2. The eonstants AnI Mn2 pns are regular eonstants of the motion (the transformations are not singular) [12] [21].

*) Two dimensional harmonie oscillator, anisotropie.

Two~ther constants of the motion are:

G • aretg Sl. -

1.

arctg wq:z

PI W P2

Remark: arctg Sl. and

i(qt

+ pi> are the aetion-angle variables of the

PI

one-dimensional harmonie oseiliator.

H and A are regular constants of the motion, G not:

~I

=

LG,qd ..

Pt;~f

:&1

=

lG,p;j -

pi!~i

~~2 = LG.o~J -q2

<IM ,C>&. p~+q~.'W2

(6.1 )

Gis the generator of a transformation whieh is a contraction in the

Q2,P2 plane and an expansion in the ql,pl plane. If we write

q.

=

q.(O),

p. •

p.(O) for the starting values of tha trajeetory

1 1 1 1

and from this

<h

and similar formulas for q2 and P2

pi

+ q1 = 2i + Pl + q1

p~ + w2q~ = -2a •

P!

+ w2q~

Solutions of (6.1) exists for the range

(6.2)

be zero (if a - ao say) and so ~he transformation (6.1) becomès

singular and the trajectory exp a

G .

col (ql,Pl,q2'PZ) stops. G is

not a regular constant of the motion ([2Il). With

arctg x =

h

ln[ (t+ix)(q-ix) -1] the function G can be written as:

(6.3)

If K

=

(PI+iql)W (p2-iooqz) then both K and K* (complex eonjugatión)

are constants of the motion. With this definition (6.3) ean be written as (2iw)-1 In K/K*. We ean in stead of K and K* also choose

the real functions Kl and K2 defined by

H,A,Kl,~ do not form a closed se.t, but we can form the closure of this set with respect to l,J and x, this results in an algebra of

constants of ~he motion.

• if wal we find constants of the mot ion of even order only.

• if wa2 _{we find} H, _{A (of order two) and}

Kl - qlPlP2-Q2P! + ~qt. ~ = P!PZ-Q!P2 +

<.

qlqzPl (of order three)

af ter closure we get constants of arbitrary order.

• if 00-3 we get H and A (second order) and two fourth order constants

of the motion: Kl • Q1P2(3PI-Ql)-3qzPl(pt-3ql) >

K2

=

PIP2(pt-3qlJ + 3qlq2(3Pl-Ql}. The algebra formed af ter closure

consists of even order constants only. etc.

The closure of {H,A,Kl,K2} is, by definition, the algebra A. We can

extend this algebra A to an algebra

Ä

by adding formal power series.

Fram this algebra Ä we can select four quantities with special

commutation relations. H A 00-1 Kl [!(ql + Pl)]-

-Z- •

Kl 00-1 K2 [!(ql + Pl)]- --2-- • Kz

These quantities "fulfill the relations

LH,AJ

=

LH,Kd .. LH,K2J - 0

LKl,KzJ - 2wA ~K2,AJ • 2wKl

We cbuld have used K and K* in stead of Kl and K2 but this is not es-sential. The set {Kl,K2,A} forms an algebra isomorphic with. the

alge-bra of SU(2). Aresult analogeous to the two dimensional isotropic

oscillato~. One remark should be made. The isomorphism is alocal one:

there always is a neighbourhood of the identity (parameter ~=O) where

the isomorphism holds. For the isotropic oscillator the transformations generated by H,A,P ,M are regular everywhere in phase-space, this is

not true for the anisotropic oscillator, for certain values of ~ the

transformation (generated by H,A.Kl.K2) become singular. The symmetry groups of H (isotropic) and H (anisotropic) are therefore not global isomorphic, only local.

2. Continuous systems.

2.1 Canonical transformations.

We shall treat systems which can be described by two canonically con~

jugated functions q(~.t), p(;,t),~ is the independent space variable

and q and p are vector functions in genera!.

The Hamiltonian H now is a functional of q and p. The equation of

motion is equivalent to the hamilton equations.

(I. 1)

ö

H{q.p}

_J

d~

H(q,p,q .p ,q _{x. x.} , .... ).

X.X.

1 1 1 J

H is the Hami~tonian density of the system and is a function of

q,p,qx.'px.' etc. In general there can be n independent x-variables. 1 1

but we shall restrictourselves to n-l,2,3 which are the most natural situations. The Hamiltonian density H can depend explicitlyon; and

t, in that case H is functional of q and p and a function of t.

...

-

...

We can construct eanonical transformations q(x,t) + q(x.t) •

....

-

....

-

-p(x,t) + p(X,t), q and pare functionals of q and p.Just as in

section 1.1 the construction ean be done in two ways, with a

genera-ting funetional F (a functional of one "old" and one "new" funetion)

or via a generator of an infinitesimal'transformation, a functional of q and p.

*

Possible choises for F are:

Fl{q,q,t} , Fa{q,p,t} , Fs{p,q,t} and F4{p,P,t}. We give, for com-pleteness, the equations whieh determine the transformations.

jj - H

=

~t

Fl {q.q,t}

a

o

--r-F3{P.q.d Op.

1.

-

-!he new field variables q and p fulfill the relation

(1.2)

!he Poissonbraeket

l.

J

betweentwo funetionals 11 and G is defined in

the following way:

(1.3)

We have written here

~~Çi)

in stead of

~~

to note that the

funetio-nal derivative is dependent on ~t we shall not do that in general,

unless confusion migbt arise. !he identieal transformation ean be

generated by F2

=

J

d

~ q(~.t) p(~.t).

!he eanonical trans format ion whieh transforms theHamilton equations to the equations

oH

-!p - qt .. 0

oH

will again result in a kind of Hamilton~acobi equation. Let the

trans-formetion be generated by Fl and take

H.O.

the equations then are:

a .

-H {p,q,t} • - ätF1{q.q,t}. (l.5)

taking together:

(1.6)

*

Tbe second metbod is to use a transfo'rmetion close to the identity.

A generating functional is

f

d3

; q(;.t) p(;.t} + eG {q,P.tl. € is a smeH parameter G.will be

cal led tbe generator of the transformation.

OG P = P - e: öq

-

öG

q=q+e:5'P

Tbe trans format ion is an infinitesimal one. sa we can treat everything

up to order €. In G we mey change

p

in p. Under the transformation

above tbe first order variations in the fields q and p become ([I]).

öq = €

* -

€ lq.GJ

2.2 Invariant canonical transformations.

Under an infinitesimal transformation generated by G{q.P,t} the varia-tion in the Hamiltonian is defined as:

oH ... H{q,p,t} - H{q,p,t} (2.1)

This can be written

OB oH

OB ...

"8P

op

+

oq oq.

(2.2)

or, with (1.1) and (1.3)

(2.3)

With q and p solutions of theHamilton equations, we find for

:~

the expression dG ... dG + LG,HJ dt dt Combining (2.1)-(2.4} we get 8

!~

...

8

~~

+"8 lG,HJ ... n{q,p,t} - H{q,P.t} - H{q,P,t} + H{q,P,t} ~ dG - - - __ 8 dt ... H{q,p,t} - H{q,p,t}. (2..4)

The trans format ion is an invariant transformation if

n

is the same

H{q,p,t}

=

H{q,p,t}

H{q,p,t} = H{q,p,t}.

remark: i f G doesn't depent explicitlyon t tben H{q,p} = H{q,pL So

the value of tbe functional is constant under tbe

transfor-mation q,p ~ q,p.

We can conclude (analogeous to tbe discrete transformations): The generator of an invariant infinitesimal canonical trans format ion

(invariant variation) is a constant of the motion and vice versa.

2.3 Special transformations. Campbell-Baker~Hausdorff-formulas.

The set of functionals G{q,p,t} with the operations + and

l.J

forms

aninfinite,dimensional Lie algebra.

The structure is much more complicated tban in tbe discrete case

(section 1.3). Apart from transformations in the fields q and p only,

there are also transformations in the independent variables, so tbere is an enormeous increase of the degrees of freedom. A lattice as given in section ].5 seems not be poss,ible for tbe algebramentioned above.

Mncb of the theory given in chapter

i

can be used here, although with

tbe right modifications.

We can associate an operator with every functional G{q,p,t} in a way analogeous to that of section 1.3.

M o ·OGo

q and p are vector functions andthe dot stands for an inner product.

The operator G is acting on functionals and functions. such as q and

p (we can interprete this functions as functionals if we use a delta-function).

G'F • lG,F

J

There is an isomorphism between the algebra of functionals with

operation

L,J,

and the algebra of operators with operation

[,1.

(3.2)

Finite transformations ar~ constructed in the usual way by exponent

i-ating the infinitesimal transformation: exp a G, the exponent inter-preted as a formal power series:

~

V

generates a one-parameter subgroup of transformations exp a G.

Let F be another generator of a one-parameter subgroup of trans

forma-tions exp ~

F.

The product of two elements of the subgroup is again

a canonical transformation.

-

-exp a G • exp ~ F • exp L (3.4)

The relation between L and ehe operators G and F can be given by

~

-

~

+ :2 a2_{f3 [G,[G,li']] + . . . . (3.5)}

An expression 1ike this ean a1so be derived for the operators of

-ehapter 1. It is sometimes impossib1e to write L in c10sed form.

A variation of a special kind is the one generated by

G -

J

d3~

[BCLPa + Bf3QS +

PCLB~

qS], with

~

a 1inear differentia1

::ct CL ai aij

operator BS • BS + BS di + BS did_j + ••••

The coëfficients of ~ and Ba.

Ba

are funetions of the independent

variables. Greek indices are used for the field;eomponents ~nd latin

indices for the components of the x-variables. The summation conven-tion is used. G generates the most general inhomogeneous linear canoni-cal varia tions

(3.6)

!he trans format ion (3.61 with. all B except Ba zero is a generalized

trans lat ion in q-space (included in the Gauge transformations of q,

introduced in chapter

11.

if Ba '

0,

all other B are zero the

trans-formation is a generalized translation in p-space (Gauge trans forma-don in p).

!he transformation (3.61 with Ba - B = 0 is the most gener al linear

a

coordinate transformation (oq depends on q alone, not p, and in a

linear wayl sometimes called a point transformation [I]. A subset of

these will be cal led internal transformations or local Gauge transfor-mations (of Gauge-theory), because they change only the fieldcomponents

(do not transform the independent variables). This kind of Gauge transformations is not the same as the kind introduced in chapter 1.

a

ex •

...

In the special caSe BB and B ~ndependent of x and t the trans

forma-tion is a global Gauge transformaforma-tion. Transformaforma-tions of the type

'i '

Xi is a function of (i,t), are transformations which aan be thought

as coming from transformations of the independent ~ variable:

- .... .. ai a i

x + x + X(x). In this case is BB

«3.6»

aqual to ~B X • Exampla:

x.a gives ~qa - € x. qa, •• There are only first order derivatives,

~ x

j ~ J

so the subgroup of this kind of transformations is isomorphic with the

general covarianee group of chapter I (the general coordinate

trans-formations of the independent variabie

ilo

The generator of the

transformation

~qa _ Bai

B

+ Baij

e

+

IJ (3 q , i f3 q. ij •••

(3.7)

can be written as

. (3,S)

~ is a function of the variable Xi and the differential operator ê

x.' If we ta ka =aB 13 • \,)13 ~a nl n2. na" Xl X2. X3 0x.' or all f ' poss~ble values of nlon2.. J

n3' we have the basis of the algebra of the general covarianee group.

There is an isomorphism between generators ,G, with Poissonbracket and

-a

.

the algebra of operators BB' ThlS can be seen by taking the

Poisson-bracket between two generators Gl and Ga of the type

(3.8).

Let ~

13 and ~ be the corresponding operators, then af ter some calculation:

(3.9)

2.4 Relation between finite and infinitesimal canonical transformations.

Just as in section 1.4 we can ask for the relationship between the generating functional of a finite transformation and the generator of an infinitesimal one. Much of the problems here are similar to those of section 1.4. Oae of the problems is the non-existence of the trans format ion for certain parametervalues.

Suppose we are working with Fa{q,p,a}. If the trans format ion generated

by Fa becomes singular for a - ao we then can use F3{p.~.a} as the

ge-nerating functional in the neighbourhood of ao. Fg and F2 are related

by a Legendre transformation.

A functional F transforms under a finite transformation generated by

the generator G, according to

The Poissonbracket between two functionals can be taken with respect to any arbitrary pair of canonically conjugated variables,so

lF,GJ

_q,p

• lF,GJ- -

_q,p

In the right hand side of (4.3) F and G must be written as

F{q{q,p}, p{q,p}}, G ·analogeous.

(4.3)

Let the generator of an infinitesimal transformation be given:

G{q,p}. We ask for tbe functional F2{·p,q,cÛ. q • q{q,p,cx} and

p •

p{q,p,a}

(4.4)a

(4.4 )b

The indices of q,p and q,p will be ignored just as before, as far as possible. The formula (4.4)a is, more precisely:

l-

GJ ..

L

(oF2 ) +

I

d3~dPiL(oF2)

q, <la - da -

-op

op.

op

~

Th _e f unct~ona . l d ' erlovatloves . -=- , -Ö Ö can e lontere ange , so b ' b d

öPi op .

l -

_{q.G • (la}

J

<l _(-=-)

oF

2 ₊

I

_d:x

3~l-

_ppG

J

--=-,

oqi

op

op

OQi 0 - --=-means

-=

qi{q,P{q,p},a} I op op q.~ (4.5) (4.6)

q can be written (in a formal way) as functional of q and p but also

of q and p. The transformation q,p + q,p is supposed to be invertible.

J

3" öG Oqi + d X

oq. / -

0- / L P P q,a (4.7) d

The left hand side can be written as 0 / :aFz

ol' N

'äcl

q,VI.

The right hand side

Tn the same way can be derived ([15])

(4.8)

Combining (4.7) and (4.8) we get

oFz {-{ - }-}

aa-

=

G q q,p,a ,p + function of a (4.9)

or,

oFz = G{oFz

p}

=

G{q, OFZ}

oa

op'

oq

(4.10)

,withG{q{q,p,a},p} = G{q,p{q,p,a}

The initial condition is

-

J -

3-Fz{q,p,O} q p d x.

From this condition: p(q,p,O) = p. The expression (4.10) is again a

generalization of the Hamilton-Jacobi equation for the continuous

case, G is the generalization of the Hamiltonian and Fz (the

functio-nal we search for) is a generalization of the action functiofunctio-nal and the parameter a plays the role of t.

2.5 Example of canonical transformations and canonical transformations as realizations of finite-parameter groups.

Let us at first stage take the simp lest case with one independent x-variable and one field variabie q. so q(x.t}.

The set xn3_x' n integer ~ 0 is the generator set of the coördinate

transformations in tbe variabie x.

G

=

I

p(x) xn3 q(x) dx is the canonical realization of this set

n x

(in the continuous case). A subalgebra of the set above is

-3 • x3 • x2_{3 with commutation relations.}

x x x [x3 , x -3 ] x = 3 x [x3 , x2_{a ] x}2_a x x x [x2₃ x· -a ] x 2xa x (5.1)

This algebra is isomorphic with the algebra J3, J! of the generators

of the group SL(2). (SL(2,R) and SL(2.~». or SU(2). respectively

the group of 2x2 real (or complex) matrices with determinant I and

the unitary 2x2 matrices, det = 1.

The operators in (5.1) are therefore a realization of L(SL(2,R» ,

the isomorphism is

-3

++1

0

x. -1

:1

x3 x

++

o

-1

o

o

The one parameter subgroups are exp(-a3 ). exp (S Xd ) and

x x

exp (yx2_{a ). working on a function f(x) in the fOllowing way}

exp (-aa ) • f(x) • f(x-a) _x exp (~x3 ) • f(x) • f(x exp S)

x

exp (yxZa_x) • f(x) - f(l~x)

Tbis is a realization of L(SL(2» in one variable .~y Lie derivatives

(first order differential operators). One ean make, more general,

realizations by generalized Lie derivatives~ operators of tbe form

f(X) a + g(x). For tbe above situation we ean find a realization of x

L(SL(2» by "solving" the eommutation relations (5.1) (substituting

operators f(x) a + g(x) for -3 , xa and x2_{a ), see [3] and [16].}

x x x x J3 ++ À + x3 X J+ ++ (2À + ~)x + x2

_a

x (5.2)

Tbe genera.lized Lie-derivatives give rise to a "loeal multiplier

re-presentation" of the corresponding group (SL(2) in this case), this

will be denoted by

[TV (g) • f] (x) - v(x;g) f(gx) (5.3)

In here g is an element of the group, v(x;g) is a local multiplier

depending on x and g}. TV is the representation with V the label of

the representation. Beeause of the groupproperties:

If J.

€

L(G). the algebra of the group G, the aetion of exp a.J. on

J J J

f(x) will then be given by solving the following equations (gener al i-zed to n • x-variables).

P.(x(a». J g

=

exp a.J.

e:

G J J o = X. 1. (5.4)

in here i and 1 runs from 1 till n, j from I to N. the number of

generators of the group G.

The trans format ion (5.3) of the fields q is generated by the

genera-ting functional.

- J

~ -1 -1

F2{q,P} = dx peg x) v(g x;g) q(x)

(if gx runs through the whole integration area if g is fixed)

q(x)

=

v(x;g) q(gx)

p(x) = v(g-l_x;g) peg-lx)

g

e:

G, the group of transformations in x.

In some cases G is alocal group of transformations, defined in a

neighbourhood of the identity but not for (some) finite parameter-values.

Au eXámple~ The operator set (5.2} with À

=

-in, n integer ~ 0 and

~

=

O. A group element can be written as

+

-ex yJ3 _{exp SJ exp aJ}

_e:

_G.

We use (5.4) to compute tbe action of f(x}.

[exp yJ3 _{exp SJ+ exp aJ-f] (x)"exp(-jny) [exp SJ+ exp aJ-f] (x exp y)}

~ exp (-!ny) (1-8 x exp y)n [exp aJ-f]

(IS

exp y )

.. I. ' (_1 ) _ Q

e

(1

»n

f(x(1+il!3)

eYJ

(lr) -a

e,(-~x»

~exp 2X X ~ xp 2X exp

(-Ix -

X

a

exp

Ix

'

The matr~x representation is

exp

~X

0

I I

1

o

exp

(-ix) •

0

o ..

(1+a!3) exp

6x

-a exp (-lr)

i f we write this as

I: .

:1'

ad-be: = 1, then we find for the ae:tion on f(x) the expression

(TV (g) fJ (x) .. (bx+d)n f(~~:a)

-a

exp

h

exp (-h)

if n-o this is a frae:tional linear trans format ion in x, a non linear

realization of SL(2)

End of exampie.

If we drop the restrie:tion to first order differential operators we

e:an find other realizations of L(SL(2»:

J3 ++

-i(xa

_x + Al

J+ ++

la

2 x J ++

iX2

This is one possibiiity, however we prefer a set of anti-hermitean operators, the operator exp aJ then is a unitary operator for il reai.