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MICROCOPY RESCLUTION TEST CHART
NATIONAL BUREAU OF STANDARDS - 1963
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Orna-P- 1386
ci-F-650619-/
lüli wow
JUL 20 1966
SPACETIME AND ELEMENTARY PARTICLE GROUPS*
M. A. Melvin and R. Roskios
Oak Ridge National Laboratory
Oak Ridge, Tennessee
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*Research sponsored by the U.S. Atomic Energy Commission under contract, with the Union Carbide Corporation.
SPACETIME AND ELEMENTARY PARTICLE GROUPS
M. A. Melvin* and R. Roskies
Oak Ridge National Laboratory, Oak Ridge, Tennessee
ABSTRACT
In this paper we discuss the possibilities of deriving mass-splitting formulas from t conjectured
coupling of the spacetime symmetry associated with isolated particles, with their approximale in.
ternal symmetry. We use the theory of the hydrogenic spectrum to illustrate the extent to which one
might expect to find mass.splitting formulas by guessing a group of higher symmetry in wirich mani.
fest symmetries are embedded. We then discuss the serious problems which arise if we assume thul
mass splitting may be interpreted in terms of the existence of a supersyi'metry group G within which
spacetime and internal symmetries are coupled nontrivially, i.e., if we assume that G does not con.
tain the mass-determining spacetine group as a direct product factor. We conclude that, burring u
possible radical redefinition of mass, the explanation of mass splitting is not to be found by em-
bedding the Poincaré group into a finile parameter Lie group G
1. INTRODUCTION AND SUMMARY
The title of this paper gives the impression of greater breadth than we can hope to attain. Actually
our main theme is what light can be shed upon the possibilities of deriving mass-splitting formulas from a
conjectured coupling of the spacetime (“external") symmetry associated with isolated particles, with their
approximate internal symmetry. By a mass-splitting formula we mean one which gives the separate masses
of the various isospin and hypercharge multiplets within each of the various supermultiplets into which
elementary particles group: the pseudoscalar and vector meson and the various baryon supermultiplels.
In Section II, as a preliminary to discussing the spacetime internal-symmetry coupling idea, we de-
velop a parable from the story of the hydrogenic spectrum. This parable serves to provide perspective
and to illustrate how much and how little – in the absence of a dynamical model – one may expect to find
correct mass-splitting formulas by guessing a group of higher symmetry in which manifest symmetries are
imbedded. Such guessing goes with interpreting the mass splitting as due either:
(I) To the operation of a "symmetry-breaking" factor which breaks or reduces the higher symmetry.
Were the latter not broken it would give a single ("unsplit'') mass value for all members of a multi-
plet; or
(II) To the existence of a nontrivial coupling of spacetime and internal symmetry so that together they
form a supersymmetry group in which the spacetime group (which is the mass determining group) does
not appear as a factor in a direct product so that the mass operator is not an invariant over the super-
symmetry group. In contrast with the broken-symmetry interpretation (1) one might also call (II) the
"augmented-symmetry interpretation" of mass-splitting.
* Permanent address: Department of Physics Florida State University, Tallahassee, Florida
In Section III we discuss in some detail the problems which arise when one attempts to interpret mass
splitting as due to the existence of a supersymmetry group G within which spacetime and internal sym.
metries are coupled nontrivially. Our general view point and principal observations with regard to this ex.
ternal-internal symmetry coupling can be summarized as follows: An elementary particle is defined by its
having sharp values of certain characterizing observables: mass, charge, spin, etc. For an isolated par-
ticle all of these are left invariant under the Poincaré group, i.e., the 10-generator Lie group comprised of
the invariant Abelian subgroup of 4 spacetime translations and 6 space-space and space-time rotations. In
the past it has been assumed that each of the characterizing observables which are not associated with
the Poincaré group (i.e., neither generators, functions of generators, nor the discrete operators of space.
reflection and time reversal which generate outer automorphisms of the Poincaré group) commute with all
the elements of the Poinca re group and they themselves form a group. All such would be called "internal.
symmetry operators" and the foregoing assumption would amount to saying (1) in physical language: there
is no coupling between the external spacetime symmetry of an elementary particle and its internal sym-
metry; (2) in mathematical language: the total-symmetry group is the direct product of an internal-symmetry
group S and the Poincaré group P.
G=SⓇP.
Leaving aside here and in the following all consideration of spacetime reflections, we can state this same
assumption in the form: the Lie algebra of G is the direct sum of the Lie algebras of S and P,
6 = volo.
Recently it has been realized that, while on empirical grounds the more directly observable internal
symmetry quantities like charge and (with strong approximation) hypercharge and isospin should commute
with all elements of P, we can no longer assume that this is true for all the other intemal symmetry oper.
ators which, after all, have been introduced historically only in order to complete a neat or consistent
algebraic structure (e.g., group structure).
In general, we may expect that some internal symmetry, operations do not commute with the elements of
the Poincaré group P. Thus P does not appear as a direct product factor in the group G. This may then
lead to mass splitting.
However, the theorem of O'Raifeartaigh' (see Sect. III) shows that, if the group G is a Lie group with
a finite number of parameters and if the mass operator Puple is not to have a continuous spectrum, then
the mass operator cannot be self-adjoint.
We thus arrive at a somewhat difficult point. If indeed there is to be a positive outcome to the program
of coupling internal symmetries nontrivially to spacetime symmetries so that there is mass splitting, it apo
pears to be necessary either to generalize the supersymmetry group to be one with an infinite number of
parameters, or else to face up to a non-self-adjoint mass operator. Both of these avenues present difficult
problems of interpretation which are discussed in Sect. III.
II. PARABLE FOR SYMMETRY-BREAKING APPROACH TO MASS.SPLITTING PROBLEM
We begin with a parable – a fictional narrative intended to illustrate a point: Our narrative is about
the theory of the energy spectrum of the hydrogenic atom. The fictional element appears in that it relates
to how much one might hope to discover about the structure of mass formulas by a scheme of higher sym-
metry – without knowing a dynamical model.
The general circular structure by which a Lie algebra and its Casimir operators are related to the total.
mass operatora for a system is indicated in the Figs. 1 and 2. We notice that there are two alternative
routes (bottom arcs) by which the eigenvalue spectrum of the total-mass operator may be found. The upper
arc corresponds to the case where we have a dynamical model (this was the historical route for the hydro-
sen atom – with the model that of Bohr). The lower arc indicates the less definite course that may be
followed if the facts of observation are used to make symmetry conjectures.
The narrative about the hydrogenic spectrum can be spelled out on the basis of the schematic flow
diagram in the lower half of Fig. 3. It concerns what might have happened if there had not been the dy-
namical model due to Bohr to explain the hydrogen term spectrum - a dynamical model which had a higher
symmetry implicit in it. (Historically this symmetry was worked out only a dozen to two dozen years after
the inception of the model.) The flow lines in the lower half of Fig. 3 show how the higher symmetry
capable of "explaining" the spectrum might nave been guessed. The dynamical model might have then
been suggested on the basis of the symmetry instead of the reverse. 3
li we wish to use the hydrogenic-spectrum parable to illustrate the supersymmetry approach in con-
trast to the broken-symmetry approach, we must shift our objective: We apply the symmetry analysis to
explain not the splitting of the term (energy) levels when hydrogenic symmetry is broken by going over tu
alkali-atom symmetry, but rather to explain the existence, within a given term (multiplet) of the hydrogenic
spectrum itself, of different angular momentum values (splitting of values of M2 within a given multiplet).
Here angular momentum, rather than mass, plays the role of the quantity whose noninvariance over the
multiplet is explained by a supersymmetry – in this case the group 0, in which the angular-momentum
determining group 0, is embedded.
III. COUPLING OF SPACETIME WITH INTERNAL SYMMETRIES
These remarks are concerned with the general program of obtaining intrinsic mass splittings by em-
bedding thé Poincaré group P into a larger symmetry group G. One of the most significant results is a
theorem due to O'Raifeartaigh.' He showed that, if one embeds P into a Lie group G that has a finite
number of parameters and if the mass operator P., Pl is self-adjoint, then in a rep of G there is either no
mass splitting, or the spectrum of masses is continuous. Neither alternative is acceptable, since a con-
tinuous spectrum of masses is physically objectionable, while no mass splitting defeats the program.
This theorem pins own quite, strongly the realm where possible mass splitting might arise. Unfortunately,
that realm becomes very small, but there is still some hope.
First, one might make the group G an infinite parameter Lie group. This possibility is largely unex.
plored, but it is possible to give instances which realize such splitting.
For example, let H, be the Hilbert space of a free particle of mass m. Then on H,, we have a unitary
representation D, of P such that
D,(P)0,(P") = m, 2.1,
· where 1 is the unit matrix in H,. Let H, be the Hilbert space of a free particle with mass m,. On H,
D, (PM)0,(PM) = m, 2.1.
Now consider the Hilbert space H, OH,. The mass operator becomes
JA BI
The group V of unitary operators on H,
in the form
H, Imatrices of the form
which are unitary I contains P
10,(P) 01
O D ,(P)]
The unitary operators form a rep of themselves and so we have a rep of the group U containing P such thal,
the mass is self-adjoint and we get a nondegenerate discrete mass spectrum. But U has an infinite number
of parameters.
The second possibility which we want to examine more carefully is to take in? not self-adjoint. This
possibility was considered even before O'Raifeartaigh's theorem appeared by Ottoson, Kihlberg, and
Nilsson. We shall briefly recapitulate their results. They start with a group G which is a product of S
and P where S is an intemal-symmetry group (a product means that every element in G can be written as a
product of an element in S and an element in P). Then they impose certain conditions on commutators of
elements of S and P and show that S is an invariant subgroup of G, i.e., G = $ x P where x denotes semi.
direct product with S as the invariant subgroup.” Moreover, if S is compact or semi-simple, they show that
G may also be written as the direct product?,8
GESP,
where p' is a group isomorphic to P, but not identical to it. The elements of P' are products of elements
of P and of S. P' is an invariant subgroup and is uncoupled from S. But P' is made up of elements from
S and P and the "interaction" of elements of P, S to produce the invariant Pmight give rise to mass
splitting.
In terms of Lie algebras, the generators of P', denoted by Pi. Müx, satisfy
Pet = P-brise
Mür = Mev – by se
where the S, are generators of S, and the b's are some nonvanishing constants.
OW
A rep of G is the direct product of a rep of S and a rep of p'. We can label the rep of G by the in-
variants of the reps of S and those of P' (i.e. mass and spin of P“). The rows ure labeled by the rows of
S and those of P' (i.e. by emlé',s'), where m denotes the row in the rep of s, p,s' are the 4-momentum
and spin labels of the rows of P9. PopoM is a constant in the rep, but P. PM is nur, i.e.
PL, PM - (P + b 23.) (POM, bmp).
IPX
and so the mass involves the generators S, and thus depends on the internul quantum numbers of the pure
ticles, which is what is wanted.
But, by O'Raifeartaigh's Theorem, P.,ple cannot be self-adjoint, and therefore in general cannot be
diagonalized. We cannot ask for the eigenvectors and corresponding eigenvalues of P., PM to obtain the
physical states of definite mass. Therefore, OKN argue that one should take expectation values of P.ph
in a certain state to get the mass of that state, for a fixed 4-moinentum P. These expectation values will
not be real, in general, but OKN argue that the complex parts are related to the widths of the states, show-
ing that the particies are unstable.
This view raises several problems. First, what does a non-self-adjoint mass mean? It certainly
means that the representation of P is not unitary, since if it were unitary, P.pit would be self-adjoint.
Unitarity is concerned with the conservation of probability, i.e. if " " is not unitary, then translation
by x does not preserve probability. OKN have argued that this is not a real difficulty, because unstable
particles decay, and thus one might expect not to conserve probability under time translation. But what
about space translation? No such argument justifies its non-unitarity. Thus, space translations should
be represented by unitary operators, while time translation need not be. However, in all cases considered
so far, where ma was not Hermitian, the space translations were not unitary. Furthermore, this division
of the energy-momentum 4-vector into space displacements and time displacements is manifestly non-
covariant.
However, there is still another interpretation open. Let us recall the case of interaction of u charged
particle of mass m and charge with an external electromagnetic field A. In that case, the canonical
momentum is not Pu = mx, but
P is the canonical momentum, Put the "kinetic momentum," and (e/c)A, the "potential momentum." The
generator of space displacement is P; it is represented in quantum mechanics by - itV. Nevertheless,
the physical mass is
(u - ») (@u - a")..2.
ulo
ulu
AM) - m2
This is the Klein-Gordon equation with electromagnetic fields. The parallel in our case is striking. Un.
der the "interaction" of P with S, the cunonical momentum becomes
internul
canonical
momentuin
kinetic
momentum
momentum
P.P. - bSP
We postulate that the generator of space-lime displacements is pro which cun be taken self-udjoini, so
that conservation of probability under translations is not violuted. Furthermore, we postulute that the
mass is not popove but PP", as is the usual cuse. This seems to overcome the previous objections.
Also, note that since Mix - Muro we have "potential ungular momentum" and "potential centroidul
moment."9
The b are determined group theoreticully; they cannot be multiplied by an overall fuctor. Thus the
"coupling strength" to the internal “field" is fixed. To extend the electromugnetic analogy, une might
say the following: Introducing the group 6 - S, P, is equivalent to coupling the particles to an internal.
symmetry field," to which we ascribe energy, momentum, etc. The coupling to this field is not wrbitrary,
but takes on certain distinct values. (Zero is always u possible value.) The physical muss is due to u
combination of space-time effects and internal-symmetry effects (say isospin, strangeress, etc.).
Now we must draw attention to a couple of discouraging features which appear when one alleinpls to
carry through this program of ascribing mass splilling to internal symmetry coupling:
(1) If the bare to be nonzero, i.e. the couplings to the internal fields are nonvinishing, then Scan-
not be a compact group. In the paper of OKN, they give an example with SU(4) as the group. But with.
out saying so, they are really considering its complex extension $44), which is not compact.
(2) The notion of the mass of a state is still not properly defined. From the expression
m2 = P.PW = (pj + bS.) (P& + blePS
for fixed PM, the operator mº involves only constants and the generators of S. Thus, for a finite dimen.
sional representation of S, m2 can be considered a matrix in the vector space of the representation of S.
OKN's prescription for the mass is to take expectation values for states labeled by definite quantum num-
bers of S. But ma is a finite dimensional matrix and therefore, even if it is not diagonalizable, it has
eigenvalues. Eigenvalues seem more intrinsic than expectation values because they are invariant under
a change of basis. It turns out that all the eigenvalues of m2 are equal and, in fact, are equal to pipoM -
m'?. Thus, with eigenvalues as masses, there is again no mass splitting. It is true, nevertheless, that
it is not necessary that ma have more than one eigenvector. Thus, using eigenvalues as masses, one
cannot readily associate the corresponding states.
If, alternatively, we interpret expectation values as masses, there are other difficulties. First, the
dia gonal elements are generally complex. But the imaginary parts are usually both positive and negative,
and thus cannot readily be interpreted as widths of resonances.'° Second, diagonal elements of a matrix
are defined only when a basis is specified. What would be a natural basis? One might choose the basis
ve
in which the matrix is in triangular form – with its eigenvalues along the diagonal. But we have ulready
pointed out that the eigenvalues of the muss matrix are all equal to mº?, and this gives no muss splitting.
The only one: natural basis scems to be the one in which the operators, whose quantum numbers label
the states, are diagonal. This basis is defined only if one chooses enough operators, so thut no two
states in the same rep of S have the same quantum numbers. For example, if S were SU(3), we would have
to choose Y, T., und T? us the cefining operators. Since there is no empirical evidence for more than a
few quantum numbers, the ussignment of many of the quantum numbers in higher symmetry groups seems
very arbitrary.
In conclusion, O'Raifeartaigh's Theorem shows that if one wants to obtain mass splitting by em-
bedding the Poincaré group P into a larger Lie group G, and if the masses are the discrete eigenvalues of
the operator P.ph, then one is forced to consider infinite purameter groups i, or a non-self-adjoint mass
operator.
Suppose that the group Gi can be written as G = $ x P with Sas ir:variunt subgroup, and can also be
written as G.S P' with p' isomorphic but not identical to P. We have shown how to admit a non-self-
adjoint mass operator PPM without violating probability conservation by distinguishing the mass-deler-
mining Poincaré group P from the group of spacetime displacements P'. The generators of the latter can
always be represented by self-adjoint operators.
On the one hand, with the masses identified as the eigen values of the non-self-adjoint operator PP",
there is s!ill no mass splitting. On the other hand, with expectation values as masses, we have difficul-
ties interpreting complex masses, and the choice of basis on which to evaluate the expectation values is
not clearly defined.
In view of these difficulties, aside from a possible radical redefinition of mass (which would have to
be consistent with the operational prescriptions followed in measuring masses) we feel that the explana-
tion of mass splitting is not be found by embedding the Poincaré group into a finite parameter Lie group
G. Whether the study of infinite para meter groups will shed more light on the problem remains to be seen.
REFERENCES
'L. O'Raifeartaigh, Phys. Rev. Letlers, 14, 575 (1965).
?For a system like the nonrelativistic liydrogen atom this is just the llamillonian. In ü speciul
relativistic invariant theory the total mass operator is the rest mass of the system considered as an
isolated and therefore "free" whole. This is the same as the energy of the system as a whole, or the
Hamiltonian, in the reference frame i which the centroid of the systein is at rest. Let the interior of
the system be represented as usual by a model of particles in fields which taper off fast as one goes
outwards. Then, even if general re!ativity – which provides the larger theoretical framework - is taken
into account, provided that the dynamics of the interior is well represented by a weak field approximation,
this total mass or energy is given approximately by a sum of "potential energies" and "kinetic energies."
(By a weak field approximation we mean that the energy-stress tensor components of the field are sufe
ficiently small so that the deviation of the metric from Minkowskian is small; the “kinetic energies" ore
taken in the usual special-relativistic sense, i.e., including the rest masses.) An example of a system
in which the field does not taper off sufficiently fast to permit a conventional definition of total mass is
the "magnetic universe" (see M. A. Melvin, Phys. Rev. 139, B225 (1965)). But even here the total energy
of a test particle in the field can be shown to be given in the weak-field limit by a sum of potential energies
and kinetic energies. (See M. A. Melvin and J. S. Wallingford, "Orbits in a Magnetic Universe," paper
submitted to Journal of Mathematical Physics.)
Sive note one interesting feature of the hydrogenic spectrum theory diagrammed in Fig. 3. The original
Lie algebra generated by the angular momentum M and the perihelion vector A has a structure somewhat
different from the semidirect product structures which have been investigated for possible nontrivial
coupling of internal symmetry with Poincare group symmetry (see references 6-8). This suggests the
following investigation: Suppose we were to follow the analogy with the hydrogenic case even more
closely and make the following radical assumption: The Lie algebra (g of the supersymmetry group for
elementary particles contains, besides the Poincare'algebra generators L (R = 1, 2, ... 10), also a
set 8 of "internal" generators S. (p = 1, ..., n) which do not form an algebra but only a "quasi-invariant
set," i.e., they satisfy the commutations relations
15.9.5.,) - COPO LA IS,L81C RS,,
where the structure constants are initially arbitrary. (The remaining structure constants are the known
ones of the Poincare 'algebra) We would then say that the Lie algebra () is the "quasi-seinidirect sum"
of Sand P: 6 = o. (Here plays the role of Mand S of u in the hydrogenic case.) Question:
Under what conditions will it be possible, by taking linear combinations of S, and L, to make up two
invariant subalgebras d' and '(generators S, and L) so that (j = o ld? The answer is given by the
following Theorem: Let G be a Lie algebra containing the quasi-invariunt sel and the Poincare'ulgebra,
P, G having a quasi-semidirect sum structure ) , P. Then, if by a change of generator base elements,
G can be rewritten in the form of a direct sum'j & P of the subalgebras d' and ', whore fuo is
isomorphic 10 , then &' can only have the structure of an algebra isomorphic to P, logether with an ucklod
center of elements which commute with all S. and Pe. We omit giving the proof of this theorem here be-
cause this special type of algebraic structure which we have termud a "quasi-semidirect sum” – though it
applies to the nonrelativistic hydrogen atom - does not seem to apply to elementary particles, as indeed
the conclusion of the theoren: shows.
"We use the term "rep"' for indecomposable irreducible representations. See M. A. Melvin, Revs.
Modern Phys. ** 28, 18 (1956); 46 32, 477 (1960) especially p. 494; 4C 34, 582 (1962).
SThe follow ing argument is based on some remarks of S. Coleman in a talk “The Trouble with Rela.
vistic SU(6)" delivered at the 1965 Washington meeting of the American Physical Society.
“U. Ottoson, A. Kihlberg, and J. Nillson, Phys. Rov. 137, B658 (1965), hereinafter referred to us
OKN.
Weaker assumptions may be made which lead to the semidirect product structure and, ultimately,
direct product structure: W. D. McGlinn, Phys. Rev. Lotters, 12, 467 (1964); F. Coester, M. Humemesh,
W. D. McGlinn; Phys. Rev. 135, B451 (1964); O. W. Greenberg; Phys. Rev. 135, B1447 (1964). M. E.
Mayer, H. J. Schnitzer, E. C. G. Sudarshan, R. Acharya and M. Y. Han, Phys. Rev. 136, B888 (1964).
Starting with the weakest assumptions the strongest theorem is given by I.. Michel, Phys. Rev. 137,
B405 (1965). Assumption No. 1 is that G is composed uniquely of S and P, i.e., element by element
GES.P= P.S uniquely. Assumption No. 2 is that there exists at least one element Sn in S whose
conjugates by all elements of P lie in S. Assumption No. 3 is that S is the only invariant subgroup of
S containing Sa The semidirect product structure follows. For those Lie groups for which the three
assumptions are satisfied the Lie algebra (y must be the semidirect sum of the Lie algebras Sand Po.
8R. Roskies, Submitted to Jour. of Math. Phys. From the development in this paper it is quite
clear that only one more assumption beyond those listed in ref 7 has to be satisfied in order that it
be possible to rewrite the semidirect sum algebra (j = IP as a direct sum algebra, (y - and op.
Since is invariant it has an infinitesimal automorphism induced by every element of . If (As-
sumption No. 4) dis of the type that all its infinitesimal automorphisms are inner (e.g., at is compact
or semis imple) this implies that every representation of likewise provides a representation of P
and makes it possible to define ten linear combinations of o generators and generators
do = Px-boy (R = 1, 2, ..., 10.
Repeated indices are summed.)
which generate a Lie algebra isomorphic with and such that which was only a semidirect sum of
Jand is now a direct sum of Sand po
'M. A. Melvin, reference(ab). See especially pp. 485 and 498.
10We are indebted to H. C. Schweinler for this remark.
10
ORAL DWG. AS-4323
LIE ALGEBRA


DYNAMICAL MODEL
TOTAL MASS (OR ENERGY)
OPERATOR H AS A FUNCTION
OF DYNAMICAL VARIABLES
REPRESENTATION
THEORY AND CASIMIR
OPERATORS C. Cgo ...
H EXPRESSED AS A SPECIFIC
FUNCTION OF C,, Cg, ...
APPLICATION OF REP THEORY
TO THE DYNAMCAL MODEL
eigenvalue spectrum
of C,, Co.o.
various possible
multiplet substructures
SPECIFIC FORM OF UNSPLIT
AND SPLIT MASS SPECTRUM
HIGHER SYMMETRY MODEL
H EXPRESSED AS A GENERAL
FUNCTION OF C,, Cg....
A. Verification of the higher symmetry group
(matching the observed with theoretical
submultiplet structure of the poup)
GENERAL FORM OF UNSPLIT
AND SPLIT MASS SPECTRUM
B. Solution of the definition problem
(idenufying physically the symmetry-breaking
opontors)
Fig. 1. Guide to the Relations Between a Dynamical Model and Symmetry Models. What is illustrated here is the
symmetry-breaking approach to the description of mass-splitting. This approach is to be contrasted with that of as.
cribing mass splitting to a nontrivial coupling of spacetime symmetry (Poincaré group P of spacetime displacements)
to internal symmetry (group S). Group theoretically this conjecture takes the following form: There is a supersyam
metry group G composed of S and p in a nontrivial way, i.e., G is not a direct-produxi structure, ils Lie algebra not u
direcl-sun structure of the Lie algebras dol S und le of P; the mass operator which was a Casimir operator in the
Poincaré group, will be a subcasimir operator (i.e., Casimir operator of an embedded noninvariant subgroup) and will
therefore show a splitting within the multiplets of G and - hopefully -- within the original multiplets old
LIE ALGEBRA I
(ut infinitouimul uporulurn, or yenurulurs, ut invuriunco group of II)
REPRESENTATION TIIEORY
AND CASIMIR OPERATORS
DYNAMICAL. MODEI.
TOTAL MASS
OPERATOR
H
AS A FUNCTION
OF DYNAMICAL
VARIABLES
C.C...
N EXPRESSED
AS A SPECIFIC FUNCTION
OF C,, C
APPLICATION OF KEP THEORY
TO THE DYNAMICAL. MODEL
SPECIFIC FORM OF
MASS SPECTRUM
(Specific dependence on
one or more quantum
numbers having definite
nanges.)
(tom:lions lotimowi, liilonowi,
olc, - of the generutors such
thout C, comnutes with every 6:116:00,1000]
Rep theory of l.le wlgebru yields
olgenvuluo spectrum of C,C,, ...
(classificulton of multiplein).
Besides it gives vurious powsible
multiplet substructures (cluusi.
ficution of submultipluis uccoid.
ing lu tho uigenvulues of certain
chaructorizing operators, In thu $.
und O groups the entire set of lliure
operutors can live chosen to be thu
Curtun ulgebru genbruturs plus ull
the "subcuvimir upesulors," j.w.
Cusimis operalors uf successive
einbedded subgroups of the sume Type
und of progrossively decreasing
ordur. More thuil one choice is
possible in generul.)
HIGHER SYMMETRY MUDES,
N EXPRESSED AS A GENERAL
FUNCTION OF C, C, ...
GENERAL FORM OF MASS SPECTRUM
(Solution of the definition problem,
by favoring the choice of a particulus
canonical set of Cusimir operuton,
helpu restrict the general form of H
and hence fixes partially the dependence
on one or more quuntum numbers having
definite nanges.)
(11, upon compurison, one of
the theoreticul multiplo!
subu fructures of the higher
symmetry ulgebru agrees with
the obwerved multiplel sube
structuro, this provides:
(A) Verilication of the highur
symmetry.
(B) Solution of the latinition
problem. This means pinning
down, through their symmetry.
breaking role, which Cartan
wigebra generators and which
Cusimir und subcasimir operu.
lors should be chosen to core
respond with physicully defined
operutors.)
Fig. 2. More Details on Rolations Between a Dynomical Model and a Symmetry Model in the Symmetry-Broaking Approach to
Masseformules.
12
LIE ALGEBRA
(1,4,1-1,2,3)
for bound orbits: [WwWl ait partie IALANI icopy M
11, M) - 1 M os Aula de para,
Or, if we set: Ho-W, X,-(x + 4x)/2 .,•(W, - A)/2
(K,Kq) = it park, Iluhalu ic parh,
(KLX1-0


Gull, - * ol!
(l is u "quusi-invariant sot";
reprouentu "quasi-veinsdirect wuni"
DYNAMICAL MODEI.
HA FUNCTION OF
DYNAMICAL VARIABLES
REPRESENTATION THEORY ANI)
CASIMIR OPERATORS
Bohr orbits in a Coulomb field:
MALO
M.A.0
M-(
mp)/
<; - (M2, A4)/2
; (M^p - P
2mZo?
M
)
C, - M.A
RhZ?
H--
**M2+ APTT
For a given buund orbit with binding
energy W.-H, we sel
Rep llicory yields:
na
pod. 2
A. Raza/wa
APPLICATION OF REP THEORY TO
THE DYNAMICAL MODEL
.-R25.
Rnza
Como
(11,8 - 1,2,...)
(n = 0,1,2, ...)
Il chuiucterizing opurulurs ure Af and
Mg, we have two cases:
13
HA GENERAL
FUNCTION OF C,
We Kna)
-0,1,2,...
Cuwe (b): 6,-,.
.
Loods to
Rydberg
a Rita
Term
Formulas
HIGHER SYMMETRY MODEL
Cuw (w): C, -
NU - Kj1) US/2011)
Study of the observad spectrum and the
splitting of the term multiplets under:
Mg +-), - (1 - 1), ...,11-1), ,
1. Transition from hydrogenic spectre
to alkall spectru.
2. Application of magnetic or electric
fields. These suggest toat:
NYMI 1) (... -1:"2
(w) The hydrogenic emitter hus the
symmetry algebra
0,-0; 0; 1,0,.
and therefore hus un intrinsic rolutionwi wymmetry, which is ruducud iu uxlul
by a ma metic fiold.
(b) Since we huvo various j's belonging to w given multiplel the physic# NJ, in to
be associated with the 10), in the quusisemidirect sum ruther thun will one of
those in the direct sum. Further, since the observed torm multiplets show
Join - 0, we are in case (a). Wo then expect thut the hydrogenic temi spec.
trum depends on only one principal quantum number na. But, further, the in.
forrod Q .(1 m, A.M - 0, structure could lend un inspired gueswer tu the
Bohr model,
Fig. 3. Illustration of Symmetry-Broaking Approach to Mass-Splitting Formulas; Application of Guide Mapped out in Fig. 2
in the Case of Hydroponic Spectro. Spelled out here oro rolations betweon the dynamical model (electron in atomic Coulomb
field) and higher symmetry model based on observations of multiplet structure of spectral forms. From the point of viow of the
dynamical model the existence of the Lio algebra of invariunco operators is vory clear. The vector ongular momentum M and
tho (Koplor Rungo Lenz-Pauli) pariholion vector A - vector from the force center of the focus of the Kepler ellipse to its
• goomno trical conter - both commute with the Hamiltonian and have certain commutation relations with each other. Togother
they gonorate the Lie algebra 4. The set of three generators A does not form a subalgebra but it is invariant under conjugu.
tion by M. We call such a sot a "quasi.invariant" set and say that is the "quasi-somidirect sum" of the set A with the
(noninvariant) subalgebra i formed by the three generators M. Linear combinations enable ono to decompose the algebra l;
into the direct sum of the two invariant subalgebras X and each isomorphic to O, the Lie algebra of the three-dimensiunul
rotation group. The well known theory of rops of the threo.dimensional rotation group onables one to define immediately the
oppropriate quantum numbers and specify the dopondence of the energy W (which is involved in the scaling of A) upon these
quantum numbers. Thus one has a "group.theoretical derivation" of the Balmor formula.
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