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For the determination of the explicit form of the operators use will be made of the
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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