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ascribed, since particles are then nonstable.
To account for this fact, it is sufficient in the above mentioned considerations to
change the linear sum to the linear integral:

?R(m)g(m), (37)
R=

where the metric g(m) is concentrated on the set composed of one or more points
(depending on how many stable particles) and nonoverlapping intervals [mi , mi ].
A more expanded formulation of equation (37) has the appearance
m
R = R (m0 ) ?
s0
(38)
Ri ,
i=1


?Rsi (m)f si (m)dm, (39)
Ri =

where Rsi (m) is the space in which is realized the irreducible representation of
algebra P with mass m and spin si ; the function f si (m), nonzero only in the interval
(mi , mi ), characterizes the “smearing” (indeterminacy) of the mass of a resonance. If
in (39) we replace f si (m) by a delta function, then R, as before, will be a linear sum
of spaces Ri .
2
(i)
The operator P? in Ri is determined in the following manner:

2 2
? P? ?m2 Rsi (m)f si (m)dm.
(i) (i)
Rsi (m)f si (m)dm = (1 )
P? Ri =

(i) (i)
The operators P? , Mµ? , M 2 , P 2 can be determined by an analogous method.
A more detailed presentation of results obtained by taking account of “smearing”
of the resonances will be given in another paper.

1. Fushchych W.I., Ukr. Fiz. Zh., 1967, 12, 741.
2. Bogolyubov N.N., Shirkov D.V., Introduction to the Theory of Quantized Fields, Wiley, 1959.
3. Fushchych W.I., Ukr. Fiz. Zh., 1967, 12, 1331.
4. Neiman I., Mathematical Bases of Quantum Mechanics, Nauka, 1964 (in Russian).
5. Barut A., Phys. Rev., 1965, 139, 1433;
Malkin I.A., Man’ko V.I., 1965, 2, 5, 230, Sov. Phys. – JETP. Lett., 1965, 2, 146.
6. Hwa R., Nuyts J., Phys. Rev., 1966, 145, 1188.
7. Fushchych W.I., Ukr. Fiz. Zh., 1966, 11, 907.
8. Foldy L., Phys., Rev., 1956, 102, 568.
9. Shirokov Yu.M., Zh. Eksp. Teor. Fiz., 1957, 33, 1196.
10. Riss F., SekefaI’vi-Nad’ B., Lectures on Functional Analysis, IL, 1954 (in Russian).
11. Jordan T., Mukunda N., Phys. Rev., 1963, 132, 1842.
78 W.I. Fushchych

12. Formanek J., Czech. J. Phys. B, 1966, 16, 1;
Votruba I., Gavlichek M., Physics of High Energies and the Theory of Elementary Particles, Kiev,
Naukova Dumka, 1967, P. 330.
13. O’Raifertaigh L., Phys. Rev. Letters, 1965, 14, 575.
14. Schweber S., Introduction to the Relativistic Quantum Theory of Fields, Harper, 1961.
15. Gel’fand I.M., Michlos R.A., Shapiro E.Ya., Representations of Rotation and Lorentz Groups,
Moscow, 1958.
16. Mettews P.T., Salam A., Phys. Rev., 1958, 112, 283.
W.I. Fushchych, Scientific Works 2000, Vol. 1, 79–82.

On a possible approach to the variable-mass
problem
W.I. FUSHCHYCH, I.Yu. KRIVSKY
The mass operator M is introduced as an independent dynamical variable which is
taken as the translation generator P4 of the inhomogenous De Sitter group. The classi-
fication of representations of the algebra P (1, 4) of this group is performed and the
corresponding P (1, 4) invariant equations for variable-mass particles are written out.
In this way we have succeeded, in particular, in uniting the “external” and “internal”
(SU2 ) symmetries in a non-trivial fashion.


The idea of variable mass has been considered by many authors in connection with
the mass-spectrum and unstable-particles problems (see e.g. refs. [1, 2]). Since for
the stable free particle M0 = P0 ? P 2 , it is supposed that the square of variable-mass
2 2

operator (in the presence of interaction, of course) is defined by M 2 ? P0 ? P 2 .
2

In connection with the problems mentioned, the idea seems to be attractive to
consider the rest mass as a variable M , on the same footing with the three-momen-
tum P . It is natural to realize this idea in such a way that we define the mass
operator M as an independent dynamical variable P4 like the components of three-
momentum P . By this the correspondence principle with the fixed-rest-mass theory
demands the free operators of energy, three-momentum and variable mass to satisfy
the condition

P0 = P 2 + M 2 ? P 2 + P 4
2 2
(1)

in this case too. It is obvious that such a definition of the mass operator is more
general than the above and is non-trivial (see below) even in the case of absence of
interaction.
The presence of a dynamical variable M ? P4 in the p-representation in addition
to the three-momentum P makes it inevitable to introduce in the x-representation (in
quantum mechanics) an additional dynamical variable besides the three coordinates x.
It is natural to take for this the canonical conjugate of M , which will be denoted below
as ? ? x4 . It should be noted in this context that at least the corresponding principle
with the fixed-rest-mass theory (not to speak of deeper physical arguments) does
not allow to consider the time t ? x0 as a dynamical variable (e.g. as canonically
conjugated to P0 ) in the variable mass theory too. Further, if we do not want to violate
the conventional connection between the momentum and configuration spaces, the
variable-mass concept discussed here requires to study the group of transformations
which conserve the five-dimensional form x2 ? t2 ? x2 ? ? 2 ? x2 , µ = 0, 1, 2, 3, 4, i.e.,
µ
the inhomogenous De Sitter group, the algebra of which we call P (1, 4), in analogy
to the algebra P (1, 3) of the Poincar? group.
e
To write P (1, 4)-invariant equations for free particles with variable mass, we make
use of the classification of the irreducible representations of P (1, 4). Analogously to
Nuclear Physics B, 1968, 7, P. 79–82.
80 W.I. Fushchych, Yu.I. Krivsky

the case of the algebra P (1, 3) (ref. [3]), we consider four classes of representations
which correspond to the values P 2 = ?2 > 0, P 2 = 0 and P 2 = ?? 2 < 0 of the
invariant

P 2 = P0 ? P 2 ? P 4 ? Pµ ,
2 2 2
(2)
µ = 0, 1, 2, 3, 4.

The algebra P (1, 4) which is determined by the generators Pµ and Mµ? has,
besides eq.(2), two other invariants:
1 12
V ? ? Mµ? wµ? , W? (3)
w,
2 µ?
4
where
1
wµ? ? ? ?µ???? M?? P? . (4)
2
For class I (when P 2 = ?2 > 0) , in the system P = P4 = 0, we have
?2 ?1
?
S 2 ? P0 W + 2P0 V = (M + R)2 = s(s + 1)? (5)
1,
?2 ?1
?
I 2 ? P0 W ? 2P0 V = (M ? R)2 = I(I + 1)? (6)
1,

where s, I = 0, 1 , 1, . . . and
2

M ? (M23 , M31 , M12 ), R ? (M14 , M24 , M34 ). (7)

It follows from eqs.(5) and (6) that in the case I all the representations of P (1, 4)
are unitarity and finite-dimensional (with respect to s and I) and are labelled by two
numbers s and I. These symbols are naturally to be identified with spin and isospin
of the free particle with variable mass m, and, owing to p2 ? ?2 , one can understand
0
the parameter ? (which is the boundary value of energy) as a “bare rest mass” of this
particle.
The P (1, 4)-invariant (in the Foldy [4] sence) equation for the wave function
?(x) ? ?(t, x, x4 ) of such a particle (and antiparticle) with arbitrary s and I has the
form
?
? 1 0
P 2 + P4 + x2 ? i
2 (8)
? ?s3 I3 (t, x, x4 ) = 0, ?= ,
??
0 1
?t

where ?s ? s3 ? s, ?I ? I3 ? I.
Thus, the variable-mass concept discussed has made it possible to unite non-
trivially the “external” (P (1, 3)) and “internal” (SU2 ) symmetries which was not
successfully done by the conventional approach to the variable mass [2].
For the class II (when P 2 = 0), in the system P1 = P2 = P4 = 0 we have
?1 ?2
P3 V = ?M P , P3 W = P 2 , (9)
?1
where Pi ? M0i + Mi4 P0 P4 , i = 1, 2, 3.
The generators P and M are those of the algebra P (3) which are evoked by
the group of translations and rotations in 3-dimentional Euclidean space. Since the
spectrum of W is continuous in this case, its values are, obviously, difficult to interpret
On a possible approach to the variable-mass problem 81

physically in an acceptable fashion. If we put W = 0 when V = 0 ven if M = 0; an
additional invariant

W ? M 2 = s(s + 1)? (10)
1

appears, so that in this case all the representations are unitary and finite dimensional
and are labelled by the spin s = 0, 1 , 1, . . ..
2
The P (1, 4)-invariant equation for the wave function of the particle (and antipar-
ticle) with variable mass m and arbitrary spin s has the form
?
P 2 + P4 ? i
2 (11)
? ?s3 (t, x, x4 ) = 0.
?t
For the class III (when P 2 = ?? 2 < 0), in the system P0 = P = 0 we have

V = ±?M N , W = ? 2 (N 2 ? M 2 ), (12)

where N ? (M01 , M02 , M03 ). The generators M and N are those of the algebra
O(1, 3) of the homogenous Lorentz group and V , W given in (12) are its invariants.
Therefore, according to ref. [5], in this case all the unitary irreducible representation
of P (1, 4) are infinite dimensional and are labelled by the numbers ?, l0 and l1 , where
? is arbitrary real, ?1 ? l1 ? 1 if l0 = 0 and l1 is imaginary of l0 = 1 , 1, . . ..
2
In this case the P (1, 4)-invariant equation for the wave function has the form
?
P 2 + P4 ? ? 2 ? i ?ll3l1 (t, x, x4 ) = 0,
l0
2 (13)
?
?t
where l?l0 = 0, 1, 2, . . ., ?l ? l3 ? l. We have got in this way, the infinite-dimensional
equation for the wave function. The physical sence of the numbers ?, l0 , l1 , l3 is not
as clear as in the former cases.
Note, by the way, that recently the infinite-dimensional equations have been inten-
sively discussed [6] (though the physical arguments underlying them are still rather
poor). Following the authors [6], one can try to find some physical sense for the
numbers ?, l0 , l1 , l3 . However, it seems that the cases I and II can more directly be
related to the problems of mass spectrum and unstable systems than the case III.
For the sake of completeness we mention that in the case IV when P0 = P = P4 =
0, we have V = W = 0. The generators Mµ? remained are those of the algebra O(1, 4)
of the homogenous De Sitter group, all representations of which are well known (see
e.g. ref. [7]) and the corresponding equations are written out in ref. [8].
Here we have written down the P (1, 4)-invariant equations for variable mass free
particles, more exactly, for elementary systems with respect to P (1, 4) (but, of course,
non-elementary with respect to P (1, 3)). The construction of the quantum field theory
on the basis of these equations and the introduction of interaction, in the framework of
the Lagrange formalism, are performed by total analogy with the conventional theory.
However, as a first step to the problems of mass spectrum and unstable systems it is
expedient to introduce some interactions in the quantum mechanical equations and to
study the corresponding models which is the subject of the next publications.
82 W.I. Fushchych, Yu.I. Krivsky

1. Mathews P.T., Salam A., Phys. Rev., 1958, 112, 283;
Lurcat F., Strongly decaying particles and relativistic invariance, Preprint, Orsay, 1968.
2. O’Raifeartaigh L., Phys. Rev. Letters, 1965, 14, 575;
Fushchych W.I., Ukrainian Phys. J., 1968, 13, 362.
3. Wigner E.P., Ann. Math., 1939, 40, 149;
Shirokov Yu.M., JETP (Sov. Phys.), 1957, 33, 1196.
4. Foldy L., Phys. Rev., 1956, 102, 568.
5. Gelfand I.M., Minlos R.A., Shapiro Z.Ya., Representations of the rotation and Lorentz Groups and
their applications, Moscow, Fizmathgiz, 1958 (in Russian).
6. Fronsdal C., Phys. Rev., 1967, 156, 1665;
Nambu Y., Phys. Rev., 1967, 160, 1171;
Takabayashi T., Progr. Theor. Phys., 1967, 37, 767.
7. Newton T.D., Ann. Math., 1950, 51, 730;
Gelfand I.M., Naimark M.A., Unit?re Darstellungen der klassischen Gruppen, Akademie-Verlag,
a
Berlin, 1957.
8. Sokolik G.A., Group methods in the theory of elementary particles, Atomizdat, Moscow, 1965 (in
Russian).
W.I. Fushchych, Scientific Works 2000, Vol. 1, 83–106.

5-

.. , ..
The mass operator is determined as an independent dynamical variable related to the
generator P4 of de Sitter’s inhomogeneous group P(1, 4) in the Minkovski 5-space. The
classification of irreducible representations of the P(1, 4) algebra is given. The wave
equations invariant under the P(1, 4) group and describing the particles with arbitrary
spin and isospin are written down in the Schr?dinger–Foldy form and isospin entering

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. 18
( 122 .)



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