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Progress of Theoretical Physics, VoL 41, No.6, June 1969

The Physical Basis of Combined Symmetry Theories*)

1<. MIRMAN

Physics DejJartment, Long Island University, The Brooklyn Center Zec1?endorf CamjJus, Brooklyn, .LVew Yorl?, 11201, U.S.A.

(Received December 7, 19(8)

The proofs (such as those of O'Raifeartaigh, .lost and Segal) that it is impossible to con­struct a theory combining the Poincare and the internal symmetry are analyzed by considering which operators the elementary particles should be considered eigenvectors of. It is found that no elementary particle is an eigenstate of the Hamiltonian (or mass-squared operator) since they are eigenstates of operators which do not commute with the Hamiltonian, and since the elementary particles are single particle states and a Hamiltonian cannot be defined for single particle states .. It is concluded that the impossibility theorems have no physical relevance. The objections to Coleman's theorem are also considered.

There has been continuing interest in the question as to whether it is pos­

sible to develop theories in which elementary particles are described by repre­

sentations of a group (or algebra) which contain the Poincare group and SU(3) as subgroups.1)-3) The general consensus of opinion seems to be that the answer

has been proven to be negative, although there has been some dispute about the mathematical correctness of the proofs. Our view is that while the proofs proba­

bly are (or can be made to be) mathematically correct, they are physically ir­relevant, and that there is no reason to think that it is impossible, in principle,

to construct such theories. Of course, even if such theories are mathematically

possible it does not mean that. they are in any way relevant to elementary par­

ticle physics, but this is another question \~hich we are not concerned with

here. $ome of the reasons for our view have been presented, though perhaps somewhat implicitly, before. 4

) It seems, however, that a thorough analysis of the physics involved would be of interest, not only in connection with combined

invariance theories. but because it may help clarify some concepts of use 111

elementary particle physics. The proofs of the impossibility theorems run about as follows: the masses

of the elementary particles (which are assumed to, belong to an irreducible re­presentation of the combined group) form a discrete set. But if the mass operator

has discrete eigenvalues then it can have only a single eigenvalue in an irreduci­ble representation. Hence mass splitting cannot be obtained in this type of theory. Jost's argumene) is somewhat dIfferent, and perhaps more reasonable.

It is that the mass spectrum has at least one discrete point, the single particle

*) Work supported by a grant from Long Island University.

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The Physical Basis of Combined Symrnetry Theories 1579

with the lowest mass, and therefore at least the eigenvector describing that particle is invariant under the group so it cannot be a member of an irreducible

multiplet containing different particles with other masses.

Let us note the important point that the impossibility theorems do not require

(nor are they claimed to require) that the properties of the elementary particles

must be in any· way "invariant" under the operations of the group, or of any of its subgroups. Thus the theorems are claimed to apply to what are called "spectrum generating groups ", "non-invariance groups" or "dynamical groups".

We define an elementary particle as an entry in the compilation of Rosenfeld,5)

and in particular we concentrate our attention on members of the well defined SU(3) multiplets,· the 8 and the 10, (these form one irreducible representation

of SU(6»; for example the "33 resonance". These particles, stable, metastable

and strongly decaying are all basis vectors of the representations and are treated

the same way by SU(3), with no distinction resulting from their decay properties or lifetimes. As far as SU(3) is concerned they are all equally good particles.

Now we notice that elementary particles so defined (and this seems to agree

\vith the common definition) are eigenstates of the spin, the isotopic spin, the Casimir operators SU(3) and other such operators, all belonging to the semi­simple part of the algebra. They are not eigenstates of the Hamiltonian*) because

the Hamiltonian connects them to other states, and the off-diagonal as well as the diagonal elements of the Hamiltonian are given in the tables. In other words

the Hamiltonian operator applied to the state describing the" 33 resonance" will

give a combination of the same state plus a state describing a nucleon and a pIOn. The matrix element between the original state and the first state in the

sum is called the rest mass (assuming the resonance is at rest) and, the other

matrix elements give the partial widths. Thus the total Hamiltonian H (and, therefore, the mass squared operator), which is the time translation operator,

takes the "33 resonance" state into a nucleon plus pion state, and therefore

does not commute with those semisimple operators, such as spin and isotopic spin, describing the properties of single particle states. This means H cannot

be diagonalized simultaneously with these operators.

So in this hypothetical combined invariance theory to find an elementary par­

ticle vve find those single particle states which diagonalize the semisimple opera­

tors; spin, isotopic spin, etc. These (from what we know experimentally and about the representations of semisimple groups) form a discrete set, with discrete

expectation values of H.

*) The impossibility theorems are usually stated in terms of the mass squared operator. How­ever, it seems more customary to think in terms of the Hamiltonian, to which the mass squared operator is equivalent in the rest system, and therefore, the word Hamiltonian will be used through­out this paper. However, there will be no change in the arguments if "mass squared" is substituted for" Hamiltonian". Of course, when we use the word" Hamiltonian" we are thinking of the par­ticle (or other system) as being in its rest frame.

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1580 R. Mirman

Note that the choice of the operators to be diagonalized is central; the objection to the proofs is that they take the wrong operator diagonal, the mass squared, or Hamiltonian. The spectrum of the total Hamiltonian, in agreement with the proofs, is continuous (except for the point due to the lowest lying particle which we discuss below). But consider any fixed energy. The eigenstate of the Hamiltonian (or to put it another way the state which in its rest system has this mass) consists of a sum of states describing the tails of various res­

onances, states of a proton plus one pion, plus two pions, etc., plus a kaon,

plus two kaons, etc. In other words this state consists of single and multiparticle states, of states of all different spin and isotopic spin, etc. This eigenstate of

the Hamiltonian is not what we mean by an elementary particle. Note that all strongly interacting particles, including stable ones, are embedded in this con­

tinuous spectrum, except the pion.

At this point a review of the hydrogen atom may be useful. By the states

of the hydrogen atom we usually mean the states labeled by the eigenvalues of the operator giving the spin of the subsystem consisting of the eleCtron and the

proton, the principal quantum number n (which is an "internal" quantum number)

and the quantum numbers describing the photons. Ignoring the photons, these

states form a discrete set, with discrete expectation values of the total Hamil­tonian I-I, but are not eigenstates of H. The eigenstates of H, which are mixtures of states of different n and spin values, *) do form a continuous spectrum.' For

the hydrogen atom there is also a "Hamiltonian" I-Io describing the electron­

proton subsystem and one may ask whether it might be possible to define a similar I-Io for elementary particle physics which we would want to have a dis­crete spectrum, and then use the impossibility theorems to show that this ele­

mentary particle Ho cannot be a generator of a combined group. The answer is no, because in making the Poincare group a subgroup of the combined group

we automatically introduce the time translation operator,. which is one of the generators of the Poincare group. This operator is the total Hamiltonian, and

all theorems must refer to it. As in the case of the hydrogen atom this total

Hamiltonian does not have a discrete spectrum, which is what we would expect from the proofs. The approximate Ho for both the hydrogen atom and the ele­mentary particles cannot be identified with the time translation operator. If the

hydrogen atom (for which we know that an Ho exists) were describable by a combined group, we would expect I-Io to belong to the semisimple part; and not the Abelian part of the group. In this sense it is not a real Hamiltonian. The

requirement that the total Hamiltonian H have a continuous spectrum gives a

condition that we know to be true experiment~l1y, except that there is one

*) Because n and the spin do not commute with the total Hamiltonian their eigenstates have a distribution of energies. That is excited states have iinite widths. So if we take the electron­proton subsystem with a given energy there is a probability of finding any of the various n anel spin values.

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The Physical Basis of Combined Symmetry Theories 1581

discrete point due to a single particle of lowest mass, and according to lost's proof this state should be invariant under the group.

A basic assumption of the impossibility theorems, including lost's, is that

the elementary particles are the basis vectors of an irreducible representation

of a combined group, and since the masses of the particles differ the diagonal

matrix elements of H differ. But, as we saw above we really cannot use one

particle states as the basis vectors of the representation because most change

into multiparticle states with times. So vve face the question of what are the

basis vectors, and in fact what is the system that H is the Hamiltonian of. It seems meaningless to define H just for the metastable particles, especially in

view of the remarks made above about the basis vectors of SU(3). First we must clarify what we mean by one particle and multiparticle states.

Vve do this in the following way, which appears plausible, although it is probably

not the only reasonable way. We assume that there exists.in the group a gauge

operator N whose eigenvalue is, by definition, the total number of particles (an

antiparticle is considered a particle here). Since N does not commute with 1-1, we fix the time so that the number of particles has a meaning. V,! e assume that

we can consider representations which, at this fixed tim'e, have only states which

are eigenstates of lV with eigenvalue of one; that is representations containing

only a sirigle particle, and that among these are representations containing the

pion, the kaon, etc., and possible other particles that do not exist within our

Universe. Note that these particles· are in different representations, not in the same irreducible representation.

In analogy with U(2) where there is a spin one representation which is a

product of two spin 1/2 representations, we define two particle states as products of two one particle states, etc., for various multiparticle states.

In our Universe it shall be assumed. that under "certain conditions" (in~

cluding those which obtain during. experiments) we can, to a good approxima­tion, at a fixed time, write the basis states as products of one particle states

with other states, which we shall call the background.

Returning now to the system we are considering, which is perhaps the whole

Universe, we take it to be described by an irreducible representation of the

combined group whose different states correspond to different states of the system, and all of whose states correspond to ,a single eigenvalue of If. We assume

that it is sufficient for the present purposes to consider only those states which can be factored into single particle states times background. By elementary

particles we mean these 'Single particle states; they all belong to a single ir­

reducible representation of the combined group, in the sense that any elementary particle is obtained by taking the single particle component of some state of the

representation. To go from one elementary particle to another, we go from one single particle state times background to a different single particle state times

background. The background must be included, in the formal discussion, because

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1582 R. Mirman

we cannot meaningfully consider the representation space to be made up of single

particle states.

The spin J (spin and J are symbolic for spin, its z component, isotopic

spin and all the other complete set of semisimple quantum number operators)

can be written as a sum of a single particle spin operator (the operator defined

over the single particle states, which gives their spin) and a background spin operator, JS + J iJ

, and these two operators, in the approximation of factoring the

states, can be considered to commute. It follows that the commutation relations of the various components of JS are the same as that of the various components

of J. Thus the single particle spin operators form an algebra isomorphic to

that of the semisimple operators. The physical reason that the two sets of

operators (at least those describing rotations) commute is that the background is to

a good approximation spherically symmetric, and we can rotate a single particle,

without rotating the background, or without producing any effect on the particle.

The elementary particles defined here are the eigenvectors of J/, etc., and

are changed into one another by J_ s, etc. The entire set of elementary particles

is the entire set of eigenvectors of J/, etc., in this representation.

Note that EI and J/ do not commute since a single particle state such as

the "33 resonance" changes its quantum numbers with time. Since Hand Jz are generators of a (presumed) known algebra, that of the combined group, we

have relations between them. However, we have no such relations between H and J/, and in order to compute the matrix elements of H, we have to assume something about them, as, for example, that they are generators of some Lie

algebra. The value of the assumption lies in its experimental. success. While we have written J as a sum of JS and Jb, we cannot write I-I in a

similar form. The reason is that H changes single particle states into multi­

particle states. If we attempted to follow the "33 resonance" as it becomes a

nucleon, for example, and tried to write HS as the operator which acts on the

single particle baryon state then we would have to consider the factorization as being time dependent, since the pion joins the background, changing it. This would mean HS and Hb do not commute. It therefore seems to be highly

doubtful that these operators would have meaning, in any reasonable approximation.

Consider the hydrogen atom, with the electric charge greatly increased, to

say the value of the strong coupling constant. While it would be reasonable to ignore the photons far from the electron-proton subsystem, it would not be

reasonable to write a Hamiltonian for this subsystem without a term giving the interaction with the radiation field. It might still be useful however to find the

eigenstates of the spin of the electron-proton subsystem, and compute expectation values of the total Hamiltonian between these states. Further if the system were described by a combined group it might be reasonable to assume different total energies correspond to different representations. Within any representation there would be a whole set of eigenstates of the spin of the electron-proton subsystem.

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The Physical Basis of Combined Symmetry Theories 1583

The basic idea behind this approach is to define eigenstates of single particle

semisimple operators, and calculate the contribution of these single particle

components to the various expectation values of H between states which are factored into single particle components and background. The single particle contributions give the masses and the partial widths but the singlle particle states

do not belong to the spectrum of H. In fact we cannot discuss a spectrum of

H because H is constant in an irreducible representation.

Another popular type of impossibility theorem is exemplified' by the paper of Coleman6

) which states that a symmetry group of the S matrix, which

contains a subgroup locally isomorphic to the Poincare group, must be locally

isomorphic to the direct product of the Poincare and internal symmetry groups.

This is an interesting theorem but it is not clear what physical application it

has. One might try to apply it to a group containing isotopic spin rotations (or SU(3)) which "approximately" commute with the S matrix. In that case

the theorem would presumably state that the group is "approximately" a direct

product of the isotopic spin group and the Poincare group. However, even this

does not seem to follow, for there is no reason to think that the operators of the group, that are not in the isotopic spin and Poincare subgroups, commute

with the S matrix, even "approximately". There seems to be no way of avoiding

the fact that the internal symmetry transformations do not commute with the S matrix, and one cannot draw rigorous conclusions by assuming that they do. In fact the concept of a group which generates mass splitting, and still commutes

with the S matrix (and probably also the concept of a nondirect product group

which commutes with the S matrix) does not seem to be well defined. Is the

scattering amplitude the same at fixed energy, momentum, velocity, speed? Thus it is not clear what is the meaning of a theorem which proves the mathematical

impossibility of achieving an undefined physical concept. What O'Raifeartaigh's theorem (which nowhere assumes anything about

operators of the algebra commuting with the S matrix) says is: that the Hamil­

tonian (or mass-squared operator) has a continuous spectrum, so that if the basis states of a representation of a group can be put into one-to-one correspondence

with the elementary particles, and if the latter are eigenstates of the Hamiltonian, they must have a continuous spectrum. The objection is that they are not

eigenstates of the Hamiltonian.

As far as we know, no difficulties for combined invariance theories have

been presented by any of the mathematical theorems. What they have proven

have been things we know from experiment; that the total Hamiltonian (to the extent that it can be defined) has a continuous spectrum, that the elementary particles are not eigenstates of it, and that there are interaetions. It seems, therefore, that the question about such theories is not whether it is in principle

impossibie to construct them, but whether they have any relevance to physics.

I would like to thank Dr. E, A. Clark for many valuable discussions.

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1584 R. Minnan

References

1) L. O'I~aifeartaigh Phys. I<.ev. Letters 14 (1965), 575; Phys. Rev. 171 (196H) , 1698. P. Roman and C. J. Koh, Nuovo Cim. 39 (1%5), 1015.

2) M. Flato and D. Sternheimcr, Phys. l'ev. LeI ters 15 (l9G5) , 9311; 16 (19GG) , 1185. :3) Res Jo::;t, Helv. Phys. Acta 39 (l9GG) , :3(i9.

1. Segal, .f. Functional Analysis 1 (19(j7) , 1.

4) 1<. Mirman, Phys. Rev. Letters 14 (1965), 6G8; 16 (l96G) , 58. 5) A. H. Rosenfeld et a!., Rev. Mod. Phys. 37 (1965), 663. 6) S. Coleman and .r. Mandula, Phys. 'Rev. 159 (1967), 1251.

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