REFERENCEIC/65/85

INTERNATIONAL ATOMIC ENERGY AGENCY

INTERNATIONAL CENTRE FOR THEORETICAL

PHYSICS

, ' S

.4

NON-COMPACT EXTENSIONSOF SYMMETRY GROUPS

P. BUDINI

t". • "

<A

1965PIAZZA OBERDAN

TRIESTE

IC/65/85

INTERNATIONAL ATOMIC ENERGY AGENCY

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

*NON-COMPACT EXTENSIONS OF SYMMETRY GROUPS

P. BUDINI

TRIESTE

3 December 1965

* Submitted to "Nuovo Cimento"

NON-COMPACT EXTENSIONS OF SYMMETRY GROUPS

INTRODUCTION

Recent attempts of relativistic generalizations of SU(6)

symmetry groups have the common feature to produce non-compact

extensions of the original groups.

This fact and the statement that the energy eigenstates of some

physical systems like atoms, molecules and nuclei may be considered(2)

as basis for the unitary representations of non-compact groups,

has given rise to the hope that these groups might play a vital role

in physics and constitute a powerful tool for exploring not only

symmetry but also dynamical properties of physical systems.

A connection between symmetry and dynamical properties of

the hydrogen atom has been shown a long time ago by

W. PAULI, V. FOCK and V. BARGMANN ^ and the possibility of

deducing mass spectra from the Casimir operator of non-compact(2)

groups (dynamical groups) has been postulated recently by BARUT

In this work we shall attempt to give a general method for

obtaining mass formula once the symmetry algebra of the system is

known and we shall apply it to simple systems.

Outline of the method.

Let us consider a given physical system at rest. Let R be

its mass operator and Aj. . . An be the generators of a simple or

semi-simple Lie algebra A of rank S. .

If

(I) [ Aj, Po ] = 0 i = 1 . . n

A is said to be the symmetry algebra of the system.

- 1 -

Where J1 J2 . . J, arpthe I Casimir operators of the algebra A.

A known theorem asser ts that Po is then a function PQ (Jx , . J ) of

the invariant of the algebra A .

Let now Bi . . . Bm be a set of generators which have the follow-

ing properties:

(A) The generators Aj . . An Bi . . . B m build up a non-compact Lie

algebra W .

(B) A is the maximal compact sub-algebra of W .

(C) The eigenstates of Po constitute a basis for the unitary

irreducible (infinite) representations of W .

The operators Bj will have non-zero matrix elements between

the different irreducible representations of the symmetry algebra A .

In this way the group W "generates" the full Pg eigenstates spectrum

of the system.

We have now two further possible conditions:

(Di) [ B i , Po ] ^ 0or

(D2) [ B i f PQ ] = 0

In the first case Po is a member of the algebra W . Let W1 . . Wg

(g > S. ) be the invariants of the algebra W , They will depend on

the generators Aj Bj Po and for every particular representation of

W they will be fixed c numbers. The equation:

(2) Wi = Wi (Aj . . An, Po , B i . . B J i= 1 . . g

will then establish the dependence of PQ on the generators AiBi .

Since the representations of W are given in terms of those of

its maximal compact subgroups A which in turn is determined by the

values of its invariants Jj . . Jf it is to be expected that the Wi -

-2-

will depend on Aj only through the invariants Ji . . J£ . (This is

immediately evident for the bi-linear invariant).

The equations (2) will still contain the generators B which

have non-zero transition elements between the different Po multiplets

and this might seem to be a difficulty in determining the function

IQ (JI • • Je ) starting from (2). But if we admit that the At appear

in (2) only in the combination Jx . . J^ and since the Wi are c

numbers for the whole spectrum, and in particular for the multiplets *

then we can deduce that the 'Bi will also appear in (2) in such a

combination as to be well determined for every multiplet subspace. *

And we will show that this is the case in the examples given. In

some cases it will happen that the representations of the physical

multiplets correspond to some of the invariants being zero. In this

case IQ will not depend on them and the degeneracy will be

correspondingly increased. This implies a correspondence between

the degeneracy of the multiplets and the number of non-zero

invariants.

This method will be applicable also in the case of broken sym-

metry. In this case PQ will not commute with some of the A±

These will then be considered as Bj and the symmetry group will

be only a subgroup of the maximal compact subgroup of W , and the

rest of the considerations will still apply. (Obviously PQ will in

general depend not only on invariants of the symmetry group but also

on some parameters defined by the Bj ).

When condition (D2) is satisfied PQ is not a member of the

algebra W which is now a non- compact symmetry algebra. In this

case PQ has the same eigenvalue for every multiplet in a given

representation of W . The whole energy spectrum is compressed

to one point for every representation of W .

Case (D2) can be considered as a limiting case of (Di) in fact

' in case that A is represented by SU(n) and W by SU(n, 'q)q < n, then it can be demonstrated (see F. Halb-

wachs - Preprint) that every Casimii operator of SU(n, c]) can be expressed as sum of terms each ofwhich is SU(n) invariant.

— o —

in general it is possible to find linear combinations of Bx to con-

stitute rising and lowering operators for 1 . The commutators

(Di) written in terms of these are proportional to the Po eigenvalue

spacing A Po (we are interested here only in the discrete spectrum).

Setting all AP^ =0 we obtain case (D2). So while it will not be

possible to obtain the mass splitting due to dynamical effects in case

(D2) the algebra W will still in this case generate the correct PQ

eigenstates spectrum.

Case {D2) comprehends those relativistic generalizations of

symmetry groups which produce non-compact groups commuting with

Po . It is to be expected that while these generalizations will give

algebras which generate the Po eigenstates spectrum they will not

in general be apt to describe dynamical properties.

We will now give two examples of application of this method to

simple physical systems.

The harmonic oscillator

It is known that the symmetry algebra A of the n- dimensional(4)

harmonic oscillator is given by SU(n) and that it generators can

be expressed in terms of the n2, bi-linear products a. ay of the

creation and annihilation operators from which the Hamiltonian

commuting with all of them is subtracted (otherwise the algebra is

U(n) ).

The mass operator of the system having SU(n) as symmetryalgebra will in general be a function of its n - 1 invariants J: . . Jn . r

But we know that in the particular case of the harmonic oscillator

the energy is defined by any one quantum number and in fact it turns

out that the irreducible representations of the harmonic oscillator

are characterized by Jj ^ 0, J2 . . Jn_2 = 0 ^4>^5\ in order to find

out the explicit dependence of E from Jj we have to find a group

- 4 .

which satisfies conditions (A), (B), (C) and (Di). Although the most

immediate non-compact extension is obtained adding to the SU(n)

algebra the Hamiltonian and the n{n + l) generators:

(3) B =

which give the algebra W of the non-compact Sp(2n) group, the

minimal non-compact algebra satisfying the requested condition is

that one of SU(n, 1) from whose invariant the energy spectrum will

be determined. The n-dimensional case will be discussed elsewhere;

let us solve here the simple but significant one-dimensional one.

The non-compact algebra is that one of the SO(1 , 2) group with(4)generators :

(4)

tj = ~ (a+ a+ + a a)

t2 - - — (a a - a a )

t3 = - (a+ a + a a+ )

The unitary irreducible representation is the infinite-dimensional one

based on the eigenstates Jn> of the operator a+a belonging to the

eigenvalues n= 0, 1, 2 . . . corresponding to the invariant

(5) t 2 2 2 = _Z3 T l Z2

2 2

From (4) one easily deduces that the non-compact part t + t2 of

the invariant J is also diagonal in the subspaces of the compact

algebra:

t l2 +

2 . i (a+ a)2 + a+ a + 1

- 5 -

Since H = 2t3nu we have finally substituting (6) in (5):

Both the eigenstates spectrum and the energy levels are determined

by the algebra and the dependence of H by a+ and a has not been

used.

The Hydrogen Atom

(3)It has long been known that the Schrbdinger theory of the

hydrogen atom possesses a symmetry represented by the group O(4).

The components of angular momentum Mi and Runge vector Ai

commute in fact with

P =

and obey the commutation relations:

[M, M ] = i M

(7) [ A , A ] = ' ^ - M

[M, A ] = iA

but do not build up a closed algebra which can be obtained only after

multiplying A by-e4u i

Hp-)

M and N do in fact build up the algebra of O(4) for 3 negative c

number (for Po positive the algebra of the Lorentz group). The

symmetry then is sui generis: valid only in the multiplet subspaces.

For the future development it is useful to project stereograph-

ically the p space on a four-dimensional sphere of radius , 2a

(3),(instead of 1 like BARGMANN v ') where p0 = S/~2MP0 ' . The co-

-6-

ordinates on the hypersphere will be:

(8)

and the generators of the infinitesimal rotations:

(9) M,, = -

in t e rms of the pL :

(10)

Po 2 Po

and the Runge vec to r :

(11) Nk = i\

The irreducible representations of O(4) are characterized by

the two invariants

F = i (M2+ N2)(12)

G = M.N

It is known that the hydrogen atom spectrum corresponds to the

representation G = 0 F = K(K+2), K + l being the total quantum

number. This representation can be based on the four-dimensional

harmonics <£k (§ ) on the unit hypersphere obtained setting n = 0

in (8).

These functions are related to the hydrogen eigenf unctions

\U (p) on p space by the relation:

(13) * 2 2

_ 5with N = 2TT (2pQ) 2 and satisfy normalizability conditions on the two

(3)spaces (see FOCK v ').

(2)In a previous work on relativistic extension of symmetry groups

it was anticipated that the hydrogen atom symmetry O(4) is a maximal

compact subgroup of a de Sitter group of symmetry. We will give

first an explicit representation of the algebra of this group satisfying

condition (Dl) which in the limit becomes the symmetry algebra

satisfying condition (D2).

We project first stereographically the four-dimensional sphere

on a five-dimensional hyperboloid; the co-ordinates on the hyper-

boloid will be:

C1 4.1

with

(15) n2 = n^n,,- n52 = 1

where c is an arbitrary constant (variation of c means a scale

transformation of the f which leaves (15) invariant).

The group of motion on the hyperboloid will satisfy the required

conditions : its generators:

(16)

define a de Sitter group with maximal compact subgroup O(4) whose

generators are given by:

- 8 -

(17)

The U5fJ can be also expressed in terms of the g and they are:

mn n - i r ° 2 + g 2 - i -( 1 8 ) ^ V 1 L 2c 3?, "

These are not yet the generators which act on the eigenfunctions

on space ? . We know in fact that since the surface elements on

the two spaces are related by

the relation between unimodular eigenfunctions in the two spaces will

be

a 9) *(*)

and the generators' action on the Y will be defined by

(20) Vpo V

which, gives unambiguously:

JNiotc that for n / 0 these generators do not commute with p ;

in fact expressed in terms of the p the (18) become (for n = 1):

- 9 -

9 P 0 J

p , P Q P2cp 0 po2 + p2 Po

In order to base the representation of the group on the sphericali

harmonic on the unit sphere we consider the particular case

^ = c =1 and the generators become:

V = U(21')

f*)These generators build up the algebra of the SO(4 , 1) group and

satisfy conditions (A), (B), (C) and (D2).

Let us examine the unitary irreducible representations of this

group. They are characterized by the invariants:

(22)

where

v 6 y

We have from THOMAS that the invariant W can be expressed in

:': Note. While this work was being completed a preprint by M. BANDER and C ITZYKSON (SLAC -DUB - 120) has appeared in which the generators (21*) of SO(4,V1) are obtained starting from the con-

formal group on l space.-10-

the form

W = G . R

where G is the biquadratic invariant of the O(4) subgroup and F

depends on V5jJ . Since G is zero in the multiplet subspaces

(23) W = 0

Correspondingly the energy depends on only one quantum number, as

it should.

According to NEWTON there are two classes of infinite

unitary representations for W = 0 : those with Q > 0 and those

with Q = - (n+l)(n+2) == 0 (n integer)and they can all be expressed

in terms of the irreducible representations of O(4) spanned by the

spherical harmonics on the £2= 1 sphere. On this space Q can

be easily calculated starting from (211) and the result is

(24) Q = f

and the irreducible representation is completely determined. The

space of the representation is the direct sum of O(4) represent-

ations subspaces,each subspace being contained at most once. For

the invariants given by (23) and (24) the space is characterized by:

(25)

Matrix elements of the generators (21') can be easily obtained with

the usual procedure

The algebra we used obeys condition (Ds) as such should not

be expected to give the energy spectrum. Nevertheless this is

a particular case of symmetry since R is contained in the sym-(3)

metry algebra O(4). We have in fact that :

-11-

(26) - » V^ VM, S 1 + 0

and it is easy to check that the non-compact part of the invariant

Q also is diagonal in every multiplet subspace as anticipated:

Substituting (27) and (26) in (22.) one obtains the Burner formula

0 2(k+i)2

This method can also be applied to the Bethe-Salpeter equation

studied by CUTKOSKY . In that case the non-compact group(2)

is the de Sitter group which enlarges to the SO(3, 2) group for

zero energy and it gives substantially the same result as that

obtained starting from requirements of relativistic general-

ization of the compact symmetry.

Conclusion

It has been shown that the non-compact extension of sym-

metry algebras are not only apt to generate the eigenstates spectrum

of the physical system but also can furnish a method for obtaining

mass formulae exact in the frame of the symmetry, the degeneracy

of the invariants being connected to the degeneracy of the multiplets.

One general condition is that the mass operator does not commute

with some of the operators of the non-compact algebra or, except-

ionally, be contained in the compact one. *

Once the generators of the algebra and the space of their

representations is explicitly given it will be possible to obtain

Green functions which will automatically take into accour

•.lie sum on intermediate states and their mass differences.

If the algebra is known and it is possible to find a space where its elements can be expressed as gener-ators of infinitesimal rotations, then, in that space, the Casimir operators will be represented bydifferential equations whose eigenvalues will furnish the mass spectra.

-12-

ACKNOWLEDGMENT

The author wishes to thank Dr. R. Racska for interesting

discussions.

-13-

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