like sources attract with an inverse square law of force. \.Jhat we see here .... here is the Newtonian law of attraction (K/87f = G), with the algebraic sense of.
"
i
Gravitons and Photons: The Methodological Unification of Source Theory
Julian Schwin ger University of California Los Angeles, Calif., 90024
I,
-2-
It was Einstein's great ambition, in his lat er y ears, to construct a unified theory of gravitation and electroma gnetism.
In thi s he failed.
The
reason for that is quite clear once the problem is considered to be a physical one, and not an exercise in pure mathematics.
There is no kno\Vll physical
phenomenon in which the gravitational action of the e le ct romagnetic field differs in any significant way from that o f any other energy-momentum bearing substance; all that counts is the distribution of the se mech anical quantities. Accordingly, if it is not to run afoul of the simplest ex perimental tests, any formal union of the gravitational and electromagnetic fields must be physically empty.
It is therefore of some interest that a uni fication, in
quite a different sense, has been achieved in rec ent years throu gh the application of source theory, a systematic phenomenologic al approach to all particle physics.
In this short essay, we shall try to give an introductory
account of that development. The problem that Einstein set himself was posed in classical field language.
Source theory uses quantum mechanical particle langua ge.
It was
inspired initially by the necessity for providing a non-sp eculative attit ude and formalism with which to describe the families of parti cl es that have been artificially created in high energy physics e xperiments.
But it was quickly
recognized that this physically motivated technique also produc ed great simplifications in the theories that are built around t wo particular particles - the photon and the graviton - the theories of ele ctroma gne tism and I.
gravitation.
Indeed, the special v a l ue s o f mass and spin (h e l icity) pos s ess ed
by these two particles, combined with the general laws of quantum mechanics and special relativity, are so restrictive that the essential fram eworks of these two fundamental theories evolve inexorably wi t h s uc h parallel lin es of rea sonin g that one can truly speak of a
mc t l 1 0d o ] o~ica l
uni f i c ~ tion.
-3-
Source theory finds an empirically secure basis for its reasoning in the idealization and abstraction of the physical manipulations used in the work of the experimentalist.
The central concept of the source emerges in an idealiza-
tion of the realistic collisions that are used to create and to detect the various types of particles.
The associated source functions serve both as a
descriptive device for characterizing the particles that enter into and emerge from the reactions of interest, and as initial models for the evolution of the detailed dynamics that is specific to each kind of particle.
Since particles
are considered to be created in order to study them, and annihilated through their eventual detection, the description of all processes begins and ends with a vacuum state (symbol: zero); this is represented by the quantum probability amplitude (wave function) S
+
-
exp[iW(S)]
Here the minus and plus tags on the vacuum symbol are causal labels, referring respectively to any time before and after the space-time region where the sources S are manipulated.
The exponential form on the right is suggested by
the existence of physically independent experimental arrangements, for which the associated probability amplitudes simply multiply and the corresponding W expressions therefore add.
The historically important connection between a
wave function and a mechanical action correctly suggests the general identification of W as this fundamental mechanical property of the system (our units are such
~hat ~
=
c
~
1). L
In the simplest situation, a particle is exchanged between a pair of sources.
We use the (physically empty) example of a massless, spinless particle
(scalar source, symbol: K) to illustrate it:
W(K)
1.2
f (dx)(dx')
K(x) D+(x-x') K(x')
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where the propagation of the particle between space-time points x and x' (note the necessary symmetry:
D+(X'-x)} is described by
D+(x-x')
dw
p
(dp) p
dw
p
e
ip(x-x')
,
o
(Zn ) 3
which is just an invariant summation of the plane waves r epresenting the possible quantum states of the particle.
We direct attention to the following
immediately derived result,
exp[-
f
dW
p
K(p)* K(p)] < 1
where K(p) is the Fourier transform of K(x), and note its consistency with the physical demand that a probability not exceed unity.
And, we point to an
arrangement where the source remains practically time-ind ependent for a long Then W basically reduces to W = -ET, where
time interval T.
E=-
i f (d~)(d~')
K(~) ~ K(~')
I
incorporates the elementary time integral 00
f
o
1
0
d(x -x ') D+(x-x')
~
-00
The physical identification of E is evident in the form of the probability ampli tude ,:
exp [-iET] ; it is an interaction energy. Two component sources, I. a and b , of dimensions that are small in comparison with their separation
distance R, have the interaction energy
k
= f
(d~)
K(x)
like sources attract with an inverse square law of force.
\.Jhat we see here
-5-
is an early, and physically incorrect, theory of gravitation (Nordstrom). We contact the real world, specifically that of photons and electromagnetism, by replacing the scalar source with a vector source, J~(x).
If we
do no more than that, as expressed by W(J)
=
tf
(dx)(dx')
J~(x)
J~(x')
D+(x-x')
with arbitrary J~(x), we encounter an unacceptable quantum mechanical difficulty:
= exp[- f can exceed unity, since positive.
J~(p)*
We cannot set J
o
dw
P
J~(p)*
J (p) ~
=
J (p)] ~
IJ(p)
-
2 1
- IJo(p)
1
2
is not necessarily
equal to zero - that is not a covariant statement. I
There must be a scalar restriction on J~(x), and there is only one possibility:
a
a
To appreciate its effect, let the spatial momentum of a particular particle
= P0
point along the third axis, so that P3
J~(p)* J (p) ~
IJl (p)[2 + IJ 2(p)
2 1
Then we have J J
l
+ iJ 21/2
2 2
J +
1
JO(p), and
3(p) - iJ
2 2
21/2
>
a
At the same time that we satisfy quantum requirements, the independent source components reduce to only two, corresponding to transverse linear or circular polarizations.
We are indeed describing the photon, a unit helicity particle.
The restriction on photon sources is a differential conservation law. I. know the name of that conserved physical property - electric charge. interaction energy of two quasi-static charge distributions is
The
We
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- I (dx)(dx')
1" J l-l(x)~, J
a
-
~TIIX-X
I
(x ") b l-l -
these are the Coulomb-Arnperian laws, which in clude th e stat ement that like charges repel with an inverse square law of forc e. The next step in the progression that began with scalar and vector sources is a symmetrical tensor source,
The same reasoning that produced the scalar restriction on th e vector source I
will demand an analogous vector restriction on this tensor source,
There is also, now, the independent possibility of constructin g a scalar,
T(x)
=
'J
T
'J
(x)
which would represept a spinless particle.
That is explicitly removed by '
writing
The basis for the particular factor, 1/2, in the se cond term becomes
transpar~nt
on evaluating
restriction p Tl-l'J(p) l-l
P3
=
= 0,
1T I2 ,
and recognizing that the source I.
applied as before in the coordinat e system where
o
P , immediately produces the reduction
Tl-l'J(p)* T (p) - 1 T(p)* T(p) 2 l-l'J
=
I
a,b=1,2
This combination is just such as to r emove the tr a ce of th e two-dimensional
l
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symmetrical tensor Ta b' thereby leaving- only two independ ent components.
The
rearrangement into
then makes explicit that we are describing a helicity two particl e - the graviton. The restriction on graviton sources is also a differ ential conservation law, and we know the name of this conserved physical prop erty - it is the mechanical attribute of energy-momentum.
Here, however, there is a conflict
of units, and we relate the graviton source tensor to the mechanical stress tensor by
where
K
units.
is necessarily positive and has dimensions of (le ngth)2 in our atomic The interaction energy of two quasi-static mechanical distributions is,
therefore, Eab
=-
KJ
(dx)(dx ') [T~V(x) a
1
~ ~
T (x') b ~V ~
For two pure mass distributions (where only
TOO
_1:. 2
T (x ) a
~
1 47f I~-~'
I Tb ( ~ ')]
I 0), that are of dimensions
negligible in comparison with their distance, this becomes K
87f here is the Newtonian law of attraction (K/87f = G), with the al gebraic sense of I,
that inverse square law uniquely determined, whereas on e must a ppe a l to experiment in the traditional Einsteinian approach. The complete interaction energy expression clearly contains more than Newtonian gravity and, indeed, it leads dire ctly to all the fnmiliar e xpe r i mentally testahle statements of general relativity.
Si n c e perihelion precession
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is the least elementary of these, we focus on it.
The interaction energy
between an arbitrary mechanical system and a mass M localiz ed at the origin (the Sun) is given by
The neglect of T and the replacement of TOO with a point distrihution of kk total mass m gives the Newtonian interaction energy V moving planet,
=-
of L\' Tkk
I~ 12
I~
I
=
-GMm/R.
For a slowly
« 1, the corrections to V include (a) the contribution (2T/m)To
T00
correction factor:
1
o,
+ (2T/m);
where T is the Newtonian kinetic energy (b) the contribution in TOO of the kinetic
energy T, supplementing the rest energy m - correction factor: 1 + (T/m); I'
(c) the contribution in TOO bf the potential energy between planet and Sun a one line calculation based on the Newtonian spatial distribution of this energy gives the additional energy:
V
2
/2m.
To these effects is added the
first relativistic correction to the kinetic ener gy, -T
2/2m,
leading to the
total perturbation
2 3T V + V I
m
2m
m
The last step here uses the non-relativistic ener gy r elation, T + V
=
E, and
the fact that a perturbation proportional to V does not produce precession. Now we have only to recall that the kinetic energy ef fect by itself
2 (-T /2m
~
2 -V /2m) implies a precession rate that is exactly 1/6 of th e
Einsteinian prediction to see that we h~,ve neatly reproduced this characteristic consequence of the theory of general relativity. An impression may have been conveyed that fields are e xorcised in source theory.
Not at all.
The field concept appears naturally on examinin g the
response of W to infinitesimal source variations.
,
The cha r nc t e r is t i c gauge
ambiguities of elec;:trotnagnetic and gravitational fields follow from the ..
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respective source restrictions.
And th e, field equation s themselves ar e
derived, straightforwardly for the Maxwell field, somewhat more elaborately for the Einstein field since there one must develop a for malism to conve y the fact that the sources of the gravitational field include th e gr av i t a ti ona l field itself, as demanded by the conservation l aws.
The po s sihility of a
geometrical interpretation of the gr av i t a t i ona l fi e ld, at th e c la ss i ca l level, emerges at this stage; it is not an input.
We con clude t hat th e source
theory approach demystifies the gr avi t a t i ona l field. special properties.
Cer t a i n ly it has v ery
But they emerge from the particular mass and he1icity
values of a particle, the graviton, not from speculations about world geometry.
References 1.
J. Schwinger, Particles, Sources, and Fields, Vol. I (Addison-Wesley, Reading, Mass., 1970).
2.
See particularly Sections 2-4 and 3-17.
J. Schwinger, Amer. Jour. Physics 42, 507 (1974), Prec ession Tests o f General Relativity - Source Theory Derivations.
Summary
. '
I
The phenomenological, non-speculative attitude of s ourc e theory is used in parallel developments of electromagnetism and gr a v i t a t i on , based on the analogous properties of the massless particles, photon and gr a v i t on , thus providing a methodological unification of these two ar eas of physics. The power and economy of the approach is, illustrat ed by an a ppl i ca t i on to perihelion precession.
·.
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Bio graphical Sket ch
Julian Schwinger is now Professor of Physics at Universit y of California, Los Angeles, after spending more than twent y-five yea r s at Harvard Universit y. He has published extensively in many areas of theore t i cal phys ics. include the Presidential Medal of Science and th e Nob e l Pri z e.
I.
Honors
