ñòð. 16 |

methods of the quantum theory of fields. The vector hi ? Ri will be presented in the

form [2]

dk Fi (k)a+ (k)|0 . (5)

hi = i

For simplicity of notation, we shall assume that the distribution function F1 (k) =

F2 (k) = F3 (k) = F (k) and all particles are without spin. The generators of the

Poincar? algebra, expressed by the operators of creation and annihilation, have the

e

form [3]

(i)

dk kj a+ (k)ai (k),

Pj = i

(i) (i) (i)

k 2 + m2 ,

dk k0 a+ (k)ai (k),

P0 = k0 =

i i

(6)

i i

(i) (i) (i) (i) (i)

?

(i)

Mlr = dk kl fr (k) kr fl (k) , M0l = dk k0 fl (k),

2 2

?a+ (k) ?ai (k)

(i)

ai (k) ? a+ (k)

i

fl (k) = .

i

?kl ?kl

We can now write the explicit form of the operators

dk dk F (k)F (k ) a+ (k)aj (k ) + a+ (k )ai (k) . (7)

dij = i j

Obviously, the operators Dij = dij Fij will transform vectors from space Ri to Rj .

Space R is irreducible with respect to operators P? , Mµ? and Dij . This statement is a

consequence of the fact that the operators Dij transform a given vector from h ? R,

to vector hi ? Ri , while, since the subspace Ri ? R is noninvariant relative to these

operators then, by the same token, the irreducibility of the representation G in R is

shown.

The set of operators (5) and (6) (and their linear combinations) form a Lie algebra

in the case where they satisfy the Jacobi identities. Calculating, for example, the

commutators [P? , Dij ]? , [P? , [P? , Dij ]? ]? etc., it is not difficult to convince oneself

that the operators derived from these are not linear combinations of the operators P? ,

Mµ? and Dij , i.e., the set G is an infinite-dimensional Lie algebra. All elements of

the algebra G can be expressed explicitly by the operators a+ (k)aj (k ), a+ (k )ai (k),

i j

+ +

[?ai (k)/?kl ]aj (k ), al (k)[?ar (k )/?ki ] and all possible products of these operators.

As will be shown below, these operators form a continuous Lie algebra. In R space

the operators of hypercharge and isospin have the form

(8)

Y = j1 F11 + j2 F22 + j3 F33 , J = i1 F11 + i2 F22 + i3 F33 ,

where j1 , j2 , j3 and i1 , i2 , i3 are hypercharges and isospins of particles m1 , m2 and

m3 . The formulas (8) permit the expression of the operator M 2 by the operator of

hypercharge and isospin. In our case

M 2 = a E + b Y + c J, (9)

70 W.I. Fushchych

where E is the unit operator, and a , b , c are arbitrary, generally speaking, constant

quantities. In the triplet representations of the algebra G, which we considered, these

quantities are uniquely determined by the masses m1 , m2 and m3 . In all other

representations, such uniqueness does not exist and hence formula (9) will give a

mass relationship between the elementary particles. If the initial particles have spin,

then

M 2 = aS + bY + cJ, (9 )

where a, b, c are arbitrary numbers and S is the spin operator.

Other examples of infinite dimensional Lie algebras, containing the algebra P , are

considered in [12].

Note 1. It is well known that the masses of elementary particles depend on spin,

hypercharge, isospin, and other quantum numbers; hence, for determining the mass

operator, one tends to express it by the operators of spin, hypercharge, and isospin. It

should be noted that, generally speaking, the mass operator can always (in principle)

be expressed by one operator. In fact, let

M 2 = f (A1 , A2 , . . . , An ), (9 )

where A1 , A2 , . . . , An are mutually commuting self-adjoint operators, operating in a

certain separable Hilbert space. In agreement with Neiman’s theorem [4] concerning

creation operators, one can determine such a bounded self-adjoint operator A in this

space that

An = ?n (A).

?

From this theorem it follows that M 2 = f (A1 , A2 , . . . , An ) = f (A), i.e., the mass

operator can always be represented as a function of only one operator A of a weakly

closed ring. This attests to the fact that there exists one universal quantum number,

with the help of which it is possible to explain the mass spectrum of elementary

?

particles if the explicit form of the function f is known.

Since the mass operator in the approach arises from the same sort of generator of

the algebra G as, say, does the operator of isospin or hypercharge, the formula (9 )

may be viewed as an equation of a hypersurface in a space of mutually commuting

operators. For such an interpretation of the mass formula (9 ), the generation opera-

tor A apparently plays the same role as does time in classical mechanics (where the

aggregate of all trajectories lies on a certain manifold, in particular on a surface

F (x, y, z) for which x = x(t), y = y(t), z = z(t)).

From the geometrical point of view the mass equations

M = a + bS(S + 1)

for hadrons and

M 2 = a2 + b2 S(S + 1)

for mesons represent “trajectories” (a parabola for hadrons and hyperbola for mesons)

of motion of the system, which can exist in various mass and spin states.

The mass equations of Okubo,

M = a + bY + c{J(J + 1) ? Y 2 /4}

A relativistically invariant mass operator 71

for hadrons and

M 2 = a + bY + c{J(J + 1) ? Y 2 /4}

for mesons, represent a hyperbolic paraboloid and double poled hyperboloid in an

imaginary three-dimensional space (M, Y, J).

In this manner, if we quantize the general equation for a hyperbolic paraboloid:

c/4y 2 ? cz 2 ? cz ? by + x ? a = 0,

i.e., if in this equation we make the substitutions x > M , y > Y , z > J, then we

will obtain the formula of Okubo for hadrons. If with each multiplet we associate

a definite hypersurface, then various transitions of one multiplet to particles of the

same multiplet can be interpreted as “motion” or the given hypersurface. Transitions

of particles of one multiplet to particles of another multiplet may be considered

as transitions from one hypersurface to another. If to all experimentally discovered

hadrons (or bosons) is assigned a single hypersurface, then all possible transitions

of hadrons (bosons) to hadrons (bosons) should be interpreted as “motion” on this

hypersurface, for which all quantum characteristics of the system can change.

2. The characteristic special feature of problems concerning the spectrum of atomic

hydrogen and of a harmonic and anharmonic oscillator, from the group theoretic

standpoint, is that all these problems can be solved by the method of embedding of

the finite dimensional Lie algebra, appropriate to groups of hidden symmetry, in a

broader but dimensionally finite Lie algebra [5, 6]. However, this statement does not

depend on where the Hamiltonian is defined — in a Hilbert or in a vector space with

indefinite metric. Thus, for example, the problem of the spectrum of an N -dimensional

oscillator with complex ghosts can also be solved by the method of embedding of a

finite dimensional Lie algebra in a finite dimensional Lie algebra2 .

From the above considerations (section 1) it follows that the Poincar? algebra

e

(relativistic case) can be included by a nontrivial method only in the infinite dimen-

sional Lie algebra (the case of non-Lie algebras are not considered here). This existing

difference between the relativistic and non-relativistic problem of the embedding of the

Lie algebra can be adequately explained in a natural manner. In quantum, mechanics,

as is well known, we always deal with finite numbers of degrees of freedom. Transition

to an infinite number of degrees of freedom, apparently, implies a transition from a

finite dimensional Lie algebra to an infinite-dimensional one. We shall expiate this

statement with an example.

As was shown in [6], the space of states of an N -dimensional harmonic oscillator

realizes an irreducible representation of the algebra U (N +1) ? U (N ). The generators

of the algebra U (N + 1) satisfy the following commutation relations:

= ??? E? ? ??? E? , ?, ?, ?, ? = 1, . . . , N + 1,

? ? ? ?

(10)

E? , E? ?

where

1

?

aµ , a+ + ,

Eµ = µ, ? = 1, . . . , N,

?

2 (11)

µ N +1

Eµ +1 = g(H)a+ ,

N

EN +1 = f (H)aµ , EN +1 = h(H),

µ

question of inclusion of an algebra of symmetry U (2N ) of such an oscillator in a dynamic algebra

2 The

will be considered in a subsequent paper.

72 W.I. Fushchych

N

a+ aµ , aµ , a+ (12)

H= = ?µ? .

?

µ ?

µ=1

If the N number of the oscillators tends toward infinity, we approach the infinite

oscillator, but then the dynamic algebra of an oscillator U (N + 1) and the algebra of

hidden symmetry U (N ) go over into the infinite dimensional Lie algebra. The algebra

Sp(2N ) may be determined by an analogous method, when N > ?.

For transition from quantum mechanics to the quantum theory of fields, it is

also necessary to let the volume in which the oscillators are “contained” approach

infinity [2]. For such passages to the limit, the operators ar and a+ are replaced

s

+

by the general operators of annihilation a(k) and creation a (k), which satisfy the

relations

= ?(k ? k ).

a(k), a+ (k ) (13)

?

The dimensionally infinite algebra U (N ) for this case is naturally associated with the

N >?

continual algebra U (k, k ), the generators of which are the operators

1

a(k), a+ (k ) (14)

E(k, k ) = .

+

2

It is not difficult to convince oneself that operators of the form (13) satisfy the

following commutative relationships:

= ?(k ? q )E(q, k ) ? ?(k ? q)E(k, q ). (15)

E(k, k ), E(q, q ) ?

Further, let us construct the algebras UN (k, k ) and Sp2N (k, k ). Consider the set of

operators:

1

?

aµ (k), a+ (k ) (16)

Eµ (k, k ) = , µ, ? = 1, . . . , N,

? +

2

E µ? (k, k ) = a+ (k)a+ (k ), (17)

Eµ? (k, k ) = aµ (k)a? (k ), µ ?

where

= ?µ? ?(k ? k ).

aµ (k), a+ (k ) (18)

?

?

Taking into account (18), it can be shown that

= ?µ? ?(k ? q )E? (q, k ) ? ??? ?(q ? k )Eµ (k, q ),

? ? ? ?

(19)

Eµ (k, k ), E? (q, q ) ?

(20)

Eµ? (k, k ), E?? (q, q ) = 0,

?

= ???? ?(k ?q )E?µ (q, k)???? ?(q?k )E?µ (q , k),(21)

?

Eµ (k, k ), E?? (q, q ) ?

= ?µ? ?(k ? q )E ?? (q, k ) + ??µ ?(k ? q)E ?? (q , k ),(22)

Eµ (k, k ), E ?? (q, q )

?

?

= ??? ?(k ? q)Eµ (k, q ) + ?µ? ?(k ? q)E? (k , q )+

Eµ? (k, k ), E ?? (q, q ) ? ?

?

(23)

+??? ?(k ? q )Eµ (k, q) + ?µ? ?(k ? q)E? (k , q),

? ?

A relativistically invariant mass operator 73

E µ? (k, k ), E ?? (q, q ) (24)

= 0.

?

?

The set of operators Eµ (k, k ) , satisfying the relations (19) form a continuous

Lie algebra UN (k, k ). The set of operators Eµ (k, k ), E µ? (q, q ) , satisfying the

?

relationships (19)–(24), form the continuous Lie algebra Sp2N (k, k ).

Utilizing the commutating relations (19)–(24) it is possible to show that the

elements from Sp2N (k, k ) ? UN (k, k ) satisfy the Jacobi identity. Since elements

of the algebra UN (k, k ) depend continuously on the variables k and k , it is then

possible to formally determine the derivative

? ?

?Eµ (k, k ) ?Eµ (k, k )

? Ai (k, k ), ? Bµ? (k, k ),

j

(25)

i, j = 1, 2, 3.

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