INSTITUTE OF THEORETICAL AND EXPERIMENTAL PHYSICS INTRODUCTION TO GAUGE THEORIES - Р -43
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«ТЕР - 4 3
INSTITUTE OF THEORETICAL
AND EXPERIMENTAL PHYSICS
LB.Okun
INTRODUCTION TO GAUGE THEORIES
M O S C O W |984;S 530.145:53^ .3 ;,i-I6
A b s t r a c t
These lecture notes contain the text of five lectures and
a Supplement. The lectures were given at the JIBR-CERN School
of Physics,' Tabor» Czechoslovakia, 5-18 June 1983. The subject
of the lectures: gauge invariance of electromagnetic and weak
interactions, hifigs^e- and eupersyametric particles. The Sup¬
plement contains^reprints (or excerpts) of some classical papers
on gauge invariance by V.Pock, P.London, O.Klein and H.Weyl,
i:: which the concept of gauge invariance was introduced and
developed.
lilli-THTyl Tri)|il'1ll'l'4Koft И .•M-r|i'|'MM4'HTIi.ll.lli>fl фи.ШКИP r e f a c e
Gauge invariant interaction* form the dynamical basis of
the modern theory of elementary particles. More than half
of 10* papers published annually in the field of particle
physics deal with gauge invariance. Some parts of the subject
are now almost as classic as Bucledian geometry.
In these lectures I will try to be as elementary as pos¬
sible, I will also try to be complementary to lectures on
gaufc < theories given at previous CERH and CERN-Dubna Schools
(B.de Wit, 1982, C.Jarlskog 1981, L.lfeiani 1980, ...) and to
the lectures given at this School. As a result I will omit
some well-known subjects and will ^ive only fragmentary
references. (A rather extensive list of references could be
/1 2/
found e.g. in books ' » ' ) • On the other hand, special
credit will be given to some old papers, selected reprints
of which are presented in the Supplement to these lectures..
Because of 1аск of space I have skipped in the written
text several topics discussed during the lectures and/or in
the transparancies:
1. Present and future experiments with W- and 2-bosons.
2. QCD and gluon-gluon collisions»
3. Grand unification and electrical neutrality of atoms.
4. Gauge fantasies on new long range forces.
I take the opportunity to express my gratitude to the
hosts of the School for their warm hospitality.C o n t e n t s Lecture 1. Gauge Inrarlance in Electromagnetic Interaction 1.1. Classical Electrodynamics of Classical Fexticlea. 1.2. Electrodynamics of Fields, 1.3. On the Etymology *nd on the Early History of Gauge Inrariance. 1,4, The Physical "•^Hng of Gauge Imrarlance In Electrodynamics. Lecture 2. Gauge Inrarlance In Weak Interaction 2.1. SU(2) Toy Model. 2.2. SU(2)xU(1) Semi-toy Eleetroweak Model, 2.3. SU(2)xU(1) ElectroweeJc Interaction. 2.4. On the Early History of Intermediate Bosons. Lecture 3» Breaking of Gauge Irrvarlance. Higgaes. 3.1. The Minimal Model. 3.2. Properties of Riggs in the Mini¬ mal Model, 3.3. Yukawa Couplings In the Minimal Model. 3.4. that is wrong with the Minimi Model? Leeture 4* Extensions 4.1. Extra Gauge Bosons. 4.2, Extra Permian*. 4.3, Extra Higgaes, 4.4* Ooldetone and Pseudogoldstone. Lecture 5. Teach-Yourself ABC of the Low Energy SUSY 5«1« Hierarchy Problem. 5.2, Superpartieles. 5.3. Vertices and Coupling Constants. 5.4. Examples of Reactions and Decays. 5.5, Example* of M A S S Pornulas. 5.6. On the Role of Graritino in the Local I«1 SUSY (Supergrarity). Supplement. Gauge Theoriest from 1919 till 1936. Reprints and Excerpts of Selected Papers by Y.Pock. ?.London. O.Klein and H.Weyl
Lecture 1. Gauge Invarlance in H <
1,1» Classical Д.ес trodtraaaics of
Let us begin with the Maxwell equations:
Vм*
where
The four-potential AM d o e e n o * appear in these equations
explicitly. It enters through the field tensor
the potential Aj* is determined up to a gradient transfor¬
mation, which is usually oalled gauge transformation, namely,
p . does not change when
wher* ЭД
Conservation of electric current is necessary for the Talldity
of the Maxwell equations' and for gajge"inrariance. it autoaati-
cally follows fron the definition of fcv г
The Maxwell equations follow from the Lagranglan
At first sight this Lagranglan is not gauge inrariant.
However the extra tent, jp.$~f $ produced by gauge trans-4
formation can be easily transformed using the current con¬
servation into a divergence: j^l^-f- -
which can be discarded if the function
has a reasonable behaviour at Infinity.
1.2. Electrodynamics of Fields
In field theory the current /•, is expressed through
charged matter fields (through wave functions of charged par¬
ticles in quantum mechanics). Consider spin 1/2 and spin 0
charged fields:
JH~ • e s fUj- v |' for Dirac field,
J •=• t€.£o>*feLCi>)- (2LCP )/«^ :
Jote the ter* 4*f / *h* *h- , whose presence is dic¬
tated by gauge invarlance.5
The "short" derivatire Э. and the four-potential AH enter
aad
the Lagrangian only through ^u О ю • Vote tfiat / C u
itself can be expressed through a commutator of long deriva-
w
tives. Consider a JL. [Q,ZX,lR'
a product JL. [Q, here £ is. аоме
(tensor)function of charged fields:
1 X
In the last expression it is implied that the derivatives aet
on Д , but not on J^ . Therefore, we can write in the operator
sense:
In field theory (and in quantum mechanics) the gauge trans¬
formation changes not only the potential of the electromagnetic
field, but the charged fields ae well. By considering the
action of the long derivative one finds that it is invariant
if the gauge transformation consists of two parts:
/ J
1,3« On the Etymology and on the Early History of
Gauge Invariance
According to Webster's Dictionary the word gauge is of Old
Norman French origin. It has several meanings:
л i a standard measure or scale of measurement,
2) dimensions, capacity, thickness etc.,
3) any device for measuring something as the thickness of
wire, the dimensions of a machined part, the amount of liquid
in a container, steam pressure, etc.Summarizing the content of the precee-dlng sections wo can add:
4) gauge in electrodynamics: a ph^ao factor which multiplies
the amplitude of a charged mailer field (or a wave function o f
e charged particle) and v/hose gradient in added to tho ele-ctro-
magnetic potential.
In other languages the corresponding wordo alno have ell
the above meanings: Eich (Gernan), jeugc (Frcnoh) ,кй.'м£Ь (R:io-
sian), «айц^ар (Serbo-Chroatl-n), Ho.i:,'•?&$dblg.iriF.n), Jcalibr
(Czech, Slovenian), kaliber (Polish, Hutch, Du::ish, Ногл\-е,~1г!л),
calibro (Itnlian), calibre (Ilinp'-'nic), T^tt (Swedish), r---rtr*i:
(Hungarian), chuan (Vietnamese), ХЭ\ iUI'3^ (.V.ancolic). I r.ppilo-
gize for being unable to reproduce Chinese and Japanese hierog¬
lyphs and giving here only their tran.vcription* f cogui"; and
[gezi"\ .
Pig. 1 gives an example of a cc-u^e; used to check the direc¬
tor of tubes. It is evident that tubec are not invariant -,.1/;.
respect to this gauge;.
Why are we using the term "gau^e invr.ritmce" in theoreti: ~,1
physics to describe a syur.cti-y which is definitely not gau^e
invariance?
The term Eichinvarianz was coined by \7eyl in 19-> 9 in the
framev/ork of his (unseccessful) uttcrapt to geometi-iao the
electromagnetic interaction tmd to construct in this way a
unified geometrical theory of gravity and eloctroruxcnctian; -
"Generalized General Relativity". At that tine Weyl used the
term Eichinvarianz аз а вупопут of scale invt»rinace ("a:;a-
Btnbinvarianz).
With tho advent of Quantum Mechanics Fock in 1926 has in¬
vented the Klein-Pock-Gordon equation (aftor Klein but before
Gordon) and discovered that tho equation in- invariant v.lth
reBpect to a tron3formation
Pock called it gradient transformation.Pig. 1.
lote tl.at if С 1* dropped» the phase factor beooaes a
scale factor. This observation was made In 1927 by London, who
thus related the phase and gradient transformation to Weyl's
old Uchtransformation.
In 1929 Weyl published a paper ("Electron and Gravitation")
In which invarianoe with respect to phase- and gradient trans¬
formation is stated as a general principle. He called it
Bichlnvarlanz•
It is Interesting that all these papers (see Supplement)
deal with the construction of unified theory of electronagne-
tiem and gravitation (Pock's paper deals with Kalnza-ELein
five-dimensional theory). The really Important and everlasting
discoveries of these papers (e.g. Weyl spinors) were considered
by the authors only a minor by-products on the way to their
lofty goal. What will survive from our grand- and super-unifi¬
cation schemes half a century later?
1.4. She Physical Keaning of Gauge Invariance
in KLectrodynamios
The physical meaning of gauge invariance in Electrodynamics
per ae does not seem at present to be terribly profound.
Por example, a tiny M*. (say ' / m « •">•* 10 1 ' cm) would des¬
troy the gauge invariance, but all our Earth-bound electrodyna¬
mics, including QED, would not be affected.Consider now the renorm&liz&bility of QED. Here gauge inva-
rience is neither necessary nor sufficient: reno: malizability
would not be destroyed by ff,v-^Q» but on the other hand» it
о•
wculd be destroyed by en anomalous magnetic moment term in the
Lagrangian, M ^ ^ w ^ f - i- u w , despite its explicit gauge ia-
variance (the dimension of Ai being twT{ ) .
What is really fundamental in electrodynamics is the conser¬
vation ot electromagnetic current: conservation of electric
charge. Without conservation of electrio charge Coulomb's law
would be impossible and photon could not be massless (see ref.
' ' and references therein). Unfortunately, the conservation of
electric charge is proved experimentallj' 1 0 1 0 times -worse than
t.ho conservation of Ъыуопас charge:
Z ( П.—•*- p + neutrals) ^ 1O 2 2 years.
Bew experiments are needed.
In textbooks on Classical Electrodynamics gauge invariance
first appeared only in 1941, in the first edition of "Field
Theory" by Landau and Lifshitz ' , which contained a special
section: Ch. 16 "Gradient Invariance". But of course the freedom
in choosing the form of the potential was exploited long before.
In this respect physicists are somewhat like the famous
Moliere's personage in Le Bourgeois Gcntilhonine who suddenly
realized that he was using prose in bis everyday conservations
all his life,
The choice of a potential is like the choice of a coordinate
/2/
syetem (see e.g. ref. ) . There is a deep analogy between the!
gauge invariance in Electrodynamics with its freedom to choose
the g^uge phase locally, and the general coordinate invariance
in General Relativity with its freedom to choose locally the
coordinate frames. In both сазез there is a sort of a J*locel
self-government".
bet U B mention here some special gauge conditions widely
used in literature:
t. •= О Lorentz gouge,
J
л — О Hamilton gauge.~ ^-J Coulomb gauge,
Ax ~ ^> axial gauge,
Ук.Д.^.0 fixed point gauge.
This last gauge was introduced by Pock '4,5/ an ^ тев ^ g ^ ел
a powerful theoretical tool by Schwinger in his book "Particles,
Sources and Pields" .» It is easy to sheck that a potential
*u = " } • J
satisfies the condition У ц Дль - O .
If a unit time-like four-vector TL and a unit space-like
four-vector § H are Introduced, then the Hamilton, Coulomb and
axial conditions can be written in a covariant form:
ТА = O (Hamilton), ^j\-fT^)(rA)^O (Coulomb),
SA » O (axial).
In many cases gauge invarlance can be used to make back-of-
the-envelope estimates of cross-sections and rates. Consider
for instance, the photon-photon scattering (Pig. 2) in the li¬
mit when the frequency of the photon &J is small compared with
the electron mass /П. : со « />7 •
By virtue of gauge irrvariance the effective Lagrangian of this
process has the fora
where o/ » 1/137 and the factor ЯгТ^ is determined by dimen¬
sional considerations ( t^l - 1 У П % It**]- /**'»*J>* Noir
let us take into account the dimension of the cross section:
Jj6^ -=«. J^ КУ>~^ "I and the fact that the cross section is proportio¬
nal to the square of the effective Lagrangian. Then from pure
dimensional arguments we get:10
where п. is a dimensionless coefficient "of the order of unity".
Lengthy QED calculations give:
(X— oc O. O G
Another example ia the de^&y JT —*-^!/ (Pig» 3 ) .
•i
Pig. 3.
The effective Lagrangian in this case is
where ^ j is the pion wave function and djj- = 130 MGV is the
famous PCAC parameter. Dimensional arguments give из the deccy
rate in the form:
r - *^/;2
where again (X. is Mof the order of unity". Accurate calculation
gives:
3QX
Kow I would like to return to the more general discussion of
the gauge invariance. One can often read in textbooks end lec¬
ture notes that local gauge invariance of the electron Lacran-
gian calls for the existence of photon field. Transformation
ч.->€'е^ ot a free electron Lngrancien produces en extra
torm «e^S--f J С ь у ^ Ч - . We need the term iCV-Ум'-рЛ
Т T T
to absorb itj* ^ Г ^ J*
Some theorists object. One can avoid introducing A*,by
giving ita role to a derivative of a scalar field CO tad trans¬
forming-- CP—»cp-*j. To be invariant under this transformation,
the'Lacfanciari cannot contain the kinetic term ( Х ф ) » a n d
the field C/> enters only thro^igh the interaction term
•f-t.£ (Эмф") ^Р/Ул4^ • H o w e v e r » because of the vector current
conservation this "interaction" is fictitioua. So the field11
Thus, only the nontrivial realization of gauge invariance call
for the existence of photons.
I do not know who first mode this observation, I heard it
from Ogievetsky, Polubarinov, Vainshtein and Khriplovich.
We can arrive at the same conclusion without introducing
the field12
where ^> - e s £
Comparing this with an abelian electromagnetic case where
S
we see that e scalar function у ie substituted now by three
scalar functions •§' , and one U(1) generator Q - by three SU(2)
generators -}r T *
How the covariant derivative Is a matrix:
The field strength is also a matrix:
In the same operator sense, as in the abelian case, ^ ^ ^ is
determined by a commutator:
Note that the last term vanishes in the abelian case but is very
important in the non-abelian case. It is trivial to see that
under gauge transformation
The Lagrangian has the form:
' and is gauge invariant.
Hote that _». _•
L This expression contains not only bilinear terms (W2) , but
, trilinear (W-') and quadrilinear (1Г) terms as well (Pig. 4).и.
"of these nonl inear self coalings I¥^he~iaiienT
feature of our toy model. It is also the salient feature of the
standard electroweek theory with its SU(2) x 0(1) gauge group.
2.2. 3U(2) x V(1) Seal-toy ELectroweak Model
bet us now take into account the fact that isotopic doublets
in Nature have nonvanishing hypercharge:
Q - I3 + Y
and consider a so to say semi-t~>y model in which the feraio-
nic doublet consists of two particles: y=» (V, -e.) , Q v « 0,
Q e .- -1, Y » -1/2.
Now not only isospin but also the hypercharge hare to be
gauged, and the gauge group is SU(2) 1 TT(1) with four gauge
fields ( W + , W", W° and B°) and two coupling constants (
and
Their ratio determine a the weak angle
wher. | V ^ - f ft
Two linear superpositions
siti
describe the photon and the 2-boson, respectively. In fact, it
is easy to check that
Henc»
where e z
The coupling of the Z-boson to a particle, that has charge Q
and isospin projection T^ isIt is Important to stress that the formulas are valid when the
symmetry SU(2) x U(1) is unbroken. They are also valid when this
symmetry is broken to ^ ( 1 ) ^ ^ .
In the case of unbrokeu SU(2) x U(1), when T, and Y are con¬
served separately, W° and B° are more meaningful then A and Z
(W° being a representation of SU(2), and B° o f U(1)). By consi¬
dering A and Z we make the first step to acknowledging that in
Nature SU(2) x TJ(1) is broken and only V(^)e/n survives. The
breaking makes W— and Z massive and only the photon stays mas-
sless.
2.3. SU(2)xU(1) Electroweak Interaction
To construct a realistic theory we have to take into account
several experimental facts:
(i)we know three generations of fundamental fermiona, of
quarks and leptona,
(ii) weak interactions are parity-violating; only left-handed
helicity states enter the charged weak currents,
(iii) quarks and at least some of the leptons are not maasless,
(iv) current quarks are superpositions of quark maas eigenstates,
(v) W and Z are heavy.
First of all let us introduce the left-handed (L) and right-han¬
ded (R) fermions;
T / p """™ ^ I '*—m /^ 5™ • f *
In accordance with (ii), let us assume that Ч»' 3 a r e SU(2)
singlets, whereas V J ' B are SU(2) doublets. There are two doub¬
lets in each generation:
*L г *TL
^ С
Here (*• ,b ,IQ are obtained from ** , Ь , v> by a unitary
transformation with four free parameters: three angles and a
phase* Hoto that because of different values of ioospin, the
L- end R- components of the same particle have different values15
of hypercharge.
ffe will defer to the next lecture the discussion of the me¬
chanism v/hich give3 зпаззсэ to gauge bosons and fermlons, I will
only explain here how the таззез of VT and Z were predicted#
Pig. 5.
Looking at the diagram (Fig% 5) describing muon decay one
0.1.0-117 finds the relation between the effective four-fermion
coupling conotanb, £| M , tho :,?•.•-•?,& £cIt)
Pig. 6.
In the standard electroweak theory g-a I ; the experimental value
cited in the Data Booklet t«, £ - 0.992+0.020. The parameter
9 т г © ^ enters the expressions for the croee шееtion» of other
neutral current reactions ae well: v V - ecattering, e//-
flcaterring, including the weak interaction of atomic electrons
with atomic nuclei, uN- scattering, and e + € ~ — • • н*Ы~ annihi-
lation. The mean experimental value of S /V» ^ ) ^ , fтот all
these processes, according to the Data Booklet, is
C'.n^ . 0.224+0.019.
This leads to the masses:
rrtw ' 77.9+U7 GeV
m-^ - 88.8+1.4 GeV.
the W- and 2-bosons were discovered in 1983 by UA1 and UA2 col¬
laborations at CERK. Professors Di Leila and Dydak will provide
you with the latest experimental data on these particles.
2.4» On the Early History of Intermediate Bosons
The non-abelian electroweak theory can be traced to several
tributaries:
1. experimental studies of weak interactions,
2. Gauge invariance of electrodynamics.
3. Spontaneous breaking of symmetries in statistical physics.
4. the conoept of lsopsin introduced by Heisenberg in 1932.
5» Yukawa's idea of mesons, the exchange of which gives both
the strong forces and the B-decay (1935).
/T/J
In 1981 Ceoilia Jaxlecog ' " has brought to general atten¬
tion a paper published by Oscar Klein in 1938. This paper in
fact developed an electroweak theory based on the isotoplc gauge
invariance.
the theory contained two doublets: p,H and Y, e. end
three vector particles: У , W* (denoted by !2>) and f~ (denoted
by o ) with gauge Invariant cubic and quartic interactions bet-17
ween them. The only coupling constant of the theory waa the
electric charge e . The possibility of neutral currente media¬
ted by the Z-boson (denoted by С ) waa also mentioned.
In building his theory Klein worked in the framework of a
fire-dimensional Kaluza-Klein world, trying to unite gravity
with electromagnetic and nuclear interactions.
Unfortunately, he did not realise that atrong and weak inter-»
action* are quite different, ao he said nothing about the valueje
of the шаввеа of В, %* and C-bosona. One ia inclined to think
that he considered theae boaona to play two rolea aimultaneoualyj:
of W+, W and Z*aad of 1Л, I "and 1°. o^ rather S + i S" «ad
S° . Furthermore, he waa not quite consistent in describing thej
electromagnetic interaction of nucleona and leptona without
introducing the hypercharge» But hie equations for the iaotopic
triplet of gauge fields are absolutely correct.
Klein's theory was firmly forgotten, and the modern non-abe-
llan theories descend from the famous paper by Yang and Kills
(1954)* Oscar Klein waa sixty whan this paper waa published. In
•953-1965 he aerred as a member of the Sobel prise committee. He'
died in 1977. Unfortunately I know nothing about Klein's reac¬
tion to the reriTal of his ideas.
I am grateful to Cecilia Jarlsoog for «ending me her paper
and a xerox copy of Klein'a paper, which is reproduced in the
Supplement to these lectures.
Lecture 3. Breaking of Oaugs Inrariance. Higgaes.
Aa we discussed in the proceeding lecture, the electroweak
gauge symmetry ia badly broken in Sature. A fundamental role in
this breaking is assigned to hypothetical spinleaa particles
called Higgs bosons or Higgses. It would perhaps be proper to
call the Higgs particle higgson, like fermion and boson. But
"higgaon" sounda aomewhat strange, like "son of Eigg * . 0 n the
other hand, a title in a reoent issue of Unclear Physics "Radi-
atire corrections to Higgs production" looks no leaa strange.
So in my lectures I will refer to the Higga boaon aa biggs,with
a lower-case h.
3.1. The Minimal Model
Let ue begin with a minimal model, containing one ieotopic
-doublet-of...scalar-bosons _ •« l^-»"t18
g£an eonfalolag -6b* field o> baa the form:
The tern l^j-tfl describe* free propagation of the field
and lte interaction with gauge fielda. Here ^ . ' ^ - I ' ^ K
-19
T72. Propertiea of
To predict /72/i we a u t know A and It It easy to find
thet
f . Indeed, ^ J L frj/fe • "°
GeT. But the ralue of J^ la unknown*
If X±m anall, XZ«oL» /??* could be cloee to it* lower lisdt
of arotmd 10 GaV. • higga with а шааа of a few tana of OaT'a
could be produoed at SP3 pp-collider end at Teratron, ассошре-
nlng the production of W e and Z'a, or at LBP in the reaction
suggeated by Ioffe and Ehose: ече~ — > ZH ,
The probability of auch higga-breuaatrahlnag ia of the order of
eereral times 10"^ for the lowest naaa raluea and rapidly de-
creaaea «hen tho ваяа of higge increase*. She main decay ohan-
nela of a light higga are
H —*- bb*. H — » - ее", H —+• 77 .
Of high interest ia the decay into two gluona, H — • - gg,
-vhich proceeds throngh Ъвепу quark loops (Tig. 7)*
Pig. 7.
A light higga ia a "cash register" of heavy quarks, its coupling
to two gluona being proportional to the nuaber of quarks q
with aaas />?,., which ia large enough; £*??- ->-n)i*% There a»y
be not only quarks In the loop*' but alao other colored heavy
particlea (quarkinoaT).
in inrerae process, gg -^ H, is an example of what ia oal-
led the gluon»gluon fuaion.The produotion of a light higga
through gluon-gluon fusion at pp- or pp-oollidera has a rather
large erosa section* 6^' v 10"^em 2 # Unfortunately, it ia dif-
fioult to dig out light higgaea produced in thla way froa under
a haystack of background erenta» It ia not ao for heary higgaea*
If Лг>(|7Г«^ • than В could be bearler taaa. W. Indeed,
r»g^ m*, at A?= i. . A higga with Г М ^ А ^ could be produced
at LIP П . The oroaa aeotion ef the proceaa e*e~ -*p-ZB (Pig.8)
ia expected to be by aa order of aegnitude saaller than the
croaa aection of the proceaa e*e~ —^-W*W~ (Pig, 9)» At U P И20
witfc гоо BeT on* expects 6Tfe+e-
the standard
croae section for the electromagnetic process e++e~e
+
U " In the lowest perturbative approximation (Pig* 10):
where is the total c m . energy.
H
Kg. 8.
Tig. 10.
It JiJi > ' > HxU then rw^ »^k,« A heavy higgs, say,
with ШЦ > 2*г>ь/ 1я difficult to produce (one needs machines
like UNK or SSC) but easy to detect. Its production will often
be accompanied by that of Z-, W-boson and it will decay into
a pair of Z- or W-bosons; thus its signature is three interme¬
diate bosons. The OAl and 0A2 experiments have revealed that the
heavier is a particle the more conspicbous are its high f> de~
cays. Three intermediate bosons in one collision would be, as
Voloshin remarked, a spectacular ^1reworks.
If Jl»/ and f^n^ TeV we will have a really strong inter¬
action in the Higgs sector and in the sector of W- and Z-bosons
(through their longitudinal components).21
ЗТЗ~ Yukawa Couplings In the Minimal Model
The same scalar doublet, which gives masses to W and Z, can
also give masses to fermions. As an example, consider quarks of
the first generation: u and d. Their right-handed conponents
Ug and dp , are isotopic singlets; iw and d, fora an isoto-
pic doublet which we denote by Q^,
The Yukawa coupling of the d^quark has the form
The Yukewa coupling of the Ux-quark involves a charge-conjugate
isopotlo doublets. (jlt. йГ
The use of Qc is necessary because of the conservation of
charge and isotopic "spin. (In the above expressions tilde ( л-> )
denotes charge-conjugate, and bar ( - ) denotes hermitian conju¬
gate of a field, the use of antisymmetrical tensor C l k allows
one to deal with antipartlcles in the same way as with particle^).
The mean vacuum value of the scalar field, «^i^^sA, gives mas¬
ses to u- and d-quarks:
In the same way the first generation of leptons acquire their
masses:
and so do the fermions of the second and the third generations.
In the framework of three generations not only diagonal mass
terms appear, like m^ULt , П7^е.е. , ^^Fh •••• hut also non-
etc
diagonal ones, like W u t uc-f »»£c CU > My,£j*-f М^Т*€ »
They originate from inter-generational Yukawa couplings of the
type of -f &kW "f/i* A e ( P • J * * s * n e s e non-diagonal masses22
tbel~are~ responsible for tbeTCobayaeM-HaeYawVliiixing" of" genera¬
tions in the charged weak currents: when mass matrix is diagona-
lized, non-diagonal current» appear* The Imaginary parts of the
non-diagonal masses are the only origin of CP-riolation in the
framework of the minimal model.
3.4. What ia wrong with the Minimal Model?
We see that aoalar fields are responsible for a large number
of fundamentally Important phenomena. Theoreticians import from
Scalarland all the masses, all the mixing angles, and CP-viola¬
tions. Therefore the discovery of Scalarland by experimentalists
should be considered as the task No.1.
At the same time the minimal model which we discussed in the
last two lectures is obviously very far from being perfect. Too
many parameters in it are arbitrary,not fixed by a priory rules*
Some of them are fixed a posteriory by experimental data, others
are still absolutely free.
In the gauge sector we have two arbitrary gauge couplings: цг
and & . The parity violation is brought into the model from the
outside by postulating that left-handed spinore form isotopio
doublets, while the right-handed ones live as singlets. The value
of the scalar condensate 4 is not predicted, the scalar self-
coupling \ and with it the hlgge mass are unknown. There is
no principle which determines the pattern of Yukawa couplings.
The lack of such a principle is especially painful in the case
of neutrino masses (not to mention the i -quark mass). We have
no reason to beleive that neutrinos are maseless. On the other
hand, we do not understand why they are so light, that is, why
their Yukawa couplings are so small.
Only experimentalists can pull Physics out of this valley of
sighs.
Lecture 4* Extensions
While awaiting new experimental discoveries theoreticians go
ahead building models, which extend the standard ••fw1?w1 mo¬
del in various directions. These models contain additional gauge
bosons, additional fermions, additional higgses. Some of the ao-
dels have broken global syanetries and Goldstonc bosons, i.e.
maseless spinless particles originating from spontaneous breaking
of. global syameJariea»- (X.will call_theaeL_perticlee. golditon* Ju.23
In tfEls "lecture we will briefly consxder some of these extended
models.
4»1. Extra Gauge Bosons
To introduce extra gauge bosons one has to enlarge the gauge
group. The beat studied example is the left-right symmetric
group (A.Salam, J.Pati, R.Marshak, R.Mohapatra, G. Senja-
novic and many others): SU(2) L x SU(2) R x U(1) ,
which contains, along with our left-handed W?, W7 and 2 T , three
extra bosons W~, Hit, Zg, which are coupled to right-handed fer-
mion doublets and left-handed fermion singlets. In the unbroken
symmetry limit the model is L-R-symmetric, and parity violation
appears in it as a result of spontaneous breaking of symmetry.
The breaking makes Wo's heavier than ^т'з, so that £*»
* чт*/*./л1л, ) «I. It also mixes W^'s and W R 's, by a small angle
С , thus producing a small coupling of light W's with right-han¬
ded fermionic currents. We know from Jb -decay that % and ^
are зта11. According to the standard model the polarization of
electrons in allowed /2>-transitions is equal to — Д> = - ' tо within -Zz.4% for P-transitions and -3^1*
for GT-transitions (van Klinken et al. ' 8 ' ) . With much better
accuracy the right-handed currents could be searched for at the
future electron-proton collider HERA, Direct search for the pro¬
duction of heavier gauge bosons is in the programs of Tevatron
and ЦИК.
4,2. Extra Fermions
Another class of models, in which parity is violated spon¬
taneously, it that of the so-called vector like models. These
models contain, along with our fermions, additional sets of the
so-called mirror fermions, which are usually assumed to be much
heavier than our ferraions. (If we assume that mirror and ordi¬
nary fermions have the same masses, then we have to conclude
that mirror fermions have their own photons, gluons, and inter-bosonsj see '*' and references therein). In the vector-
models the "sin" of parity violation could be attributed
completely to the fermion (and scalar) sector without involving
gauge bosons. Some theorists consider this as a vurtue.
4.3. Extra Higgses
There are models in which u- and d-.quarks get their masses
from two different higgs doublets. In this case there are five
physical higgses: H + , H~, H ° , H° ,- H° . The charged higgses
should be produced in pairs, electromagnetically:
e + e~ • >' У "^ H+H~«
Experimental search for this process at PETRA excludes charged
higgses with tun < 1 3 GeV. (For a recent review see ref. ' 1 ' ) .
Extra higgses should be looked for also among the decay products
of Z- and W-bosons:
^ п л , и — у а. л t п—• • ^ ' tx—ti .
Of special interest is the reaction
e*e~ >- Z -'^- V^H^ ,
where Z is virtual. The point is that the vertex ZWH vanishes in
the theories in which all scalar multiplets are doublets. The
discovery of this process would signal the existence of scalar
multiplet(s) with isospin larger than 1/2.
A neutrino can have a Dirac mass term
frivv = r
produced by Yukawa term
(feere [_i_ — ( e ) , and the scalar field (Pc has been
discussed earlier).
It can have, however, a Majorana mass. For a left-handed neutrino
the Uajorana mass term, when expressed through Weyl'a spinors,
has the form
Hermitien
^("^иУс ^^л. * conjugate)
Here c^?Jb • 1,2 are spinor indices and ^ ^ i s an antisymmetric
tensor. The Direc таэз term transforms V L into V b (see fig.
11a); the. Majorana mass. ±exm-transforms- Vl, into /Уд Xsee11b) and therefore violates conservation of leptonie charge.
Such Напогань павв term сип be produced by higgs isotopic trip¬
let with three components, , "*• 2 -
^'i is produced by the va¬
Experimentally /W^of l/e is not larger than 30 eV.(A new ITEP
experiment again gives СгЗО eV ^ a few eV# but up to now
there were no independent measurements of comparable accuracy
which could confirm or disprove this value), with л A /j*~ 10
GeV this means that "T-^^x- ^" **^ • T^ere is no explanation
why they should be во small,
Another possibility," which is actively discussed, is that у
is massless (ми-»- о ) butbut VV is
isvery
verymassive
massive ((Л9
Л 9 ^ Ю ' GeV).
V, and y are mixed by the Dirac mass /vj >r 1MeV, the mass
matrix being
гкг \
(
Then the mass of the lightest mass-eigenstate is J< ~
Majorana neutrinos trigger the neutrinoless douple & -decay
(see fig, 12) and an intricate patterm of neutrino oscillations.
*-
m,
Pig. 11.
Pig. 12.
Both phenomena are eagerly searched for, but without any posi¬
tive results. But let us end this digression on neutrino маеsee26.
and. return to scalar bosons.
With several scalar nultiplets it is possible to construct
nodela in which CP is violated not (only) through the Yukawa
coi:pliu-j3, but (also) in the hicc 3 sector. The mechanism of this
violation could bo cither explicit (complex coupling constants of-
nonlinear interactions between scalr.rs) or spontaneous. In the
latter case the Lt:cz-p.n;^ian io CP-invariant, all coupling constants
ocinc real, but the condensates - the vacuum mean values of sca¬
lar fiolda - are complex and thus CP-noninvariant.
If the only source of CP-violation i3 in the higgs sector,then
it could be shown (see ref. ' 1 '» 1 *-' ) that the dipole moment of ;
the neutron, д. > "?-з to bo of the order e.10~ -" en, v/hich is on
Vnj hrirJ: of contradiction with e>:pcrir;.ontal data. Moreover in
'•.'.-.I" cr.ee obliged hijc.-зсз have to be r-.ithyr li£ht ц
v.hLch, -•; •••-•-• hivj >IIHO acer., does not coon to be borne out by
cxpr-rir.cr.t3.
Lot чч i'-.ontion here t!:.n.t if CP is violated зроп1ялеоиз1у and
>.hc vjicuwn is characterized by a phase factor e » it is always
;v.v%ciblo to have a conplex conjugate vacuum with Q~ , That
г;огпя that vacuum domains with alternative sign3 of CP-violating"
р::^г;ол have to be considered. These domains v/ould be separated
by very thin but very maasive \vall3, which would influence cos-
r.olo£icr.l history of the universe.
a frnd Pn
The r.pinlcar, productз оГ the breaking of a local symmetry are..
called hj.-^зсэ. Iri^cc-з are masaivo. The rcplnless products of the
'.-rofikin^ of a global syiunetry are called goldstons. Goldstons are
nasriV-ss. If tha spontaneously broken symmetry is from the
bo^inninc an approximate one (broken explicitly or by quantum
ano/nnlieo) then n goldoton becomes a paeudogoldston. Pseudogold-
stons have nonvanishing masses.
An example of composite (non-eler.entary) poeudogoldston3 are .
piona, Piona are paeudocoldotons produced by the dynamical break¬
ing of c^-cbal chiral iootopic invariance of QCD. This invariance.
is an approximate one, since the u- and d-que.rks are light but
not r,;asnlccs. 'iYci-е u- and d-quarks mas3le33,pion3 also would be !
Except for pions and their SU(3)-relntivea J^- and к -mesonsл
no other (pseudo)golrtstons hare been observed experimentally.
Up to now they live only on the pages of theoretical papers, I
will mention here some of these hypothetical particlest axion,
nrion, majoron, familon, technipion ... Axion corresponds
to the global chiral UO)-symmetry in the quark sector. Arlon
/19-20/ c o r r e a p o n a 3 to the global chiral U(1)-ejmmetry in the
/21 22 23/
lepton sector. Pamilone ' • • •" correspond to the so-called
horizontal symmetry, relating fermions of different generations.
Majoron ' corresponds to the global conservation of the lep-
tonic charge. All these particles are assumed to be produced by
spontaneous breaking of the symmetries involved. Technipiona
/of* ?7/
* are pseudogoldstons that are produced by the dynamical
breaking of global isotopic chiral symmetry of a hypothetical
technicolor interaction. The interaction of a goldaton Cp with
the relevant current Jj, (the conservation of which i3 broken) is
described by an expression
J_
V
The constant V has the dimension of mass and characterizes the
scale at which the symmetry is broken.
From the experimental absence of the decay
К —*"" X + familon
one can deduce '2Л~2-' that V > 1010GeV for the global horizon¬
tal symmetry between d, s, b quarks. Somewhat lower limit for V
could be obtained from the absence of the decay
к —э»- ^. + familon
in the case of the horizontal e - M —T -symmetry.
The exchange of mnssless goldstons would lead to long-range
forces. These forces would be especially interesting in the case
of diagonal vertices that transform a fermion into itself. Such
diagonal vertices would generate low-energy long-range interac¬
tions between particles of stable natter. It is easy to see that
the vector part of the diagonal vertex } "^"^tV» ^ ,would give a
vanishing contribution because of the equation сЛ» vi"XV4+' "* 0.
The nonvanishing contribution would be given by the axial-vector
part; H'tfj-YsM--
In the static U n i t
I28
where Q is the four-momentum of the goldston {arion) and "29
If we consider the contribution to the mass of the scalar
field given by Pig. 13, we find that it is divergent. In a re-
normalizable theory there is no natural cut-off value for thio
divergence and the first place where it can be cut ('from the out-
aide) is the so-called Planck mass, /77o« , where gravitational
** 1 /2
interaction becomes strong, (As you perhaps know, /Wp/^-Grr Cr
1Q
o= 10 7 GeV, where G™ is the famous Newton constant). But in
this case we would get £«•*- fripB and Gp >~~ G H , while experimen¬
tally GJT/GJ> ~ 1 0 " ^. This extremely snail ratio, this hierarchy of
scales, is a real challenge to theorists.
—? n
н к
Pig. 13.
A possible way to avoid the above con-stop flight to the Planck
mass is an accurate compensation between the bosonic loop of Pig.
13 and the fermionic loop of Pig. 14. To achieve such compensation
one needs a sort of symmetry between bosons and fermions. Another
physical quantity which calls for boson-femion compensation ia the
so-called cosmological term - the energy-momentum tensor of the
vacuum. In this case the compensation has to be miraculously
accurate.
H /А И
• О
Pig. H .
The simplest form of boson-fermion symmetry is the so-called
N - 1 supersymmetry (N • 1 SUSY), in which there is one super-
partner for each known particle.
5.2. Superparticlea
We will use for a superpartner of a known particle suffix ino
and decorate the corresponding letter by a (super)hat or (super)
tilde. (The hat may be more convenient than tilde, as the latter
is often used in the literature to denote antiparticlea).us begin with spin-sero "inoe** leptinoe and quarkinos,
which are often reforreain the literature as sieptons and
equarks:
л
€ —»» в (einc, or electrino, or nelectron)
h -~>"J4 (ir.uino.nrcu, srauon)
У"—^- 7 s (tauino, stau)
(e-Kuino, e-cnu)
(mu-nuino, nu-snu)
(trtu-; v.ino, tau-snu)
ii*—*-u r (uino, u-squark)
d —чг d (dino, d-squnrk)
э —*r- a (air.o, s-squci'k)
с —^- о (citio, c-squark)
b —^-b (bino, b-squork)
t —^-1 (tir.o, t-scup.rk,Л'•''
'Л с rr-.i" cnp-i'"lf " А К О З " г.гг: £ol>5r*xno П (fro:i the so-called
• ;;p-r,;olrt.rtc;:o effect), hi^cino, h, (con;:ti::.o3 crilled shicg3)
and several gauginos:
photino ( V , another notation Л у )
О** • о
gluino (л A , r.nothor notation A A )
о i 3
wino ('./, other notations Дм», \ ^ )
sino (Z. nVnor notntiona J,^ , >'• ) .
Of лрсс!г1 1:T;->;:V-.::CO in ::оЛ«'"л thoorotical r.oJols is the
'I'^.ri.-ir-lurv of l!v- --r-vLloii, th'j jravitiuo, pax-ticlc with spin
Z/2, гП''П donated by Д,
]n o r " ' r to ; - тГсг:.! .-.'.г vA? о:ь.з.-..\1,'.: oi' cron^~-cction3 end
ii. .•:.-.y• г Л---С cf ;-..>-UC.-.;^3 i u v o l r r r ; ; "ino3 ! l i t ifl useful t o drrvv
•:•:;:•••....•."••':•: 3'..^ г -п ;-г-.Г" я . Него i a n c t o p l o rccipo,- how t o
•\ :•,• v- rtLCC."1 tn Mio -rr.phr:. Co:%.3iflcr en o r a i r a r y v o r t e x , say,
':•;•-.;' С 'с. - r - ) . 0\-::.; r-).\ l i n o л i d c n i i c ^ l l y (bocnurjo i n Q
: •. '. :.:.? : o::•"•"" -..:•? •Топкий" w i l l bo intorchr v !.'jcd). Crovrn
:• •: (•" I'-..- r.-'.'l,i'.:lo г--.-".гч with t a ^ ' : r h i t 3 ( P i c . 15Ъ). Кос,! tho
_"•-'.• л v.l-1-: • 1 l-'i c:a-rs . ' j j j _ . Sor.G o t h e r e x p
3
i. ';•• •* ';-.-V:bio.--;T4ir..-) v c r t i . - x ( P i ; : . 16), Л \ ' - 'l 7Г ^ ' &
Tic. 17).31
3. Emission of a goldstino by an electrino (Pig. 18).
The coupling constant is dimension!ess: з е - ( rn£ - r>\ .
where m b i s the scale of SUSY brealcinc. Accordine to the
VosuoWs W j = Ю 1 1 GeV.
If
e) b.1
Pig. 15.
a) b)
Pig. 16.
Pig.18.
5.4. Examples of Reactions and Decays
In order to draw more complex Feyrmnn diagrams, link vorti
ces,- keeping inos' lines continuous. Псге are some examples:32
Production of eluinos in gluon-gluon fusion (Pig. 19).
Note that cluino's colour charge is large. Therefore
above the,- threshold the cross section of the production of
a pair of eluinos will be much larger than that of a pair
of qunrka with the same mace:
3
3
a) b)
3
c) d)
Pig. 19.
2. The decay of gluino into photino and hadrons (Pig. 20):
л
a) ' b)
Pig. 20.
3. Interaction of photino with a quark may result in a produc¬
tion of gluino, if the energy of the photino is high enough,
Oct —>- &a (Pig. 21). The corresponding cross section ie
* 1
where
i e the raase of the (virtual) quarkino.33
з
а) Ъ)
Another possibility is the elastic scattering (Pig. 22) with
The negative results of CHARM Ъеып dunp experiment ney be
/24/
interpreted ae ' •" :
2 GeV, 1 A ^ GeV.
It is interesting to search for gluinos at SPS pp-collidcr.
If glxrinos are heavy their signature ia large nd.&sir^ v
without accompanying leptons.
A
ь г
а) Ь)
Pig. гг.
4» Decays of W (and Z) into inos, if the inos are light (Pig.
23).
A.
W e. v
V
а) Ъ) с)
Pig. 23.
л л
5. Decays of W and Z (Pig. 24)
Л Z4 A> А л
«Л
¥ — ^ - 4 + hadrons, W —*?- у • hadrons, Z — > - «> + hadrone.a) b)
Pig. 24.
л
л £• 25). V
6. Decays of У 1: у —••»—
(Pig. 26), у '»- %е~+ hadrone (Pig. 27). Let ue mention
that decay rate of the first of these processes (muino — • -
5/1,
О • 1.
goldstino + neutrino) is of the order of
V
Pig. 25.
'G. V e /e
a) b)
Pig. 26.
a) b)
Pig, 27.
5.5. Examples of Mass Formulas
Nobody really knows what the masses of ino's are. Here are
predictions from some of the recent preprints:35
A l l t h e s e formulas c o n t a i n /?"?,.. , t h e rrsr-j o^ >jr: :••: :.J. • -. .'-.
educated c u e a s :
\
Ilote tlv-t t h r - ::d.r-s bet'.voen h i c c s i r . e s ( 2! ) r-.гЛ -.'.:.:.:..::
(W and Z) ia essential:
t h e n i x i n g !!— -*.••>• V36
R e f e r e n c e s
1. Л.Б.Окунь. Лаптопы и кварки. Наука. М. 1981.
L.B.Okun. Leptone and Quarks. North-Holland, 1982.
2. А.А.Славнэв, Л.Д.Фаддеев. Введение в квантовую тэорию калиб¬
ровочных полей.
A.A.Slavnov, L.D.Paddeev. Introduction to Quantum Theory
of gauge fields. North-Holland, 1981.
3. Л.Д.Ландау, 2.М.Лифшиц. Теория поля. М. 1941.
4. З.А.Фок. Собстаанное время в классической и квантовой механи¬
ке. Известия АН СССР. Серия физ. 1937 ОМБН, с.551-568.
5. В.А.Фок. Работы до квантовой теории поля. Издательство ЛГУ,
1957, с. 150-165.
п. J.Schwinper. Particles, Sources and Fields. Addison-Wesley.
••970, v.-!, ch. 3, "И.
7. C.Jarlsc-ccg. Physica Scripta, 24, 367-872 (1981).
8. J. van Klinken et el. Phrs.ReT.Lett., £0, 94 (1983).
9. L.B.Ckun. On a search for mirror particles. Preprint ITEP-
149 (1983).
10. S.Yamada. Search for new particles. Preprint DESY 93-100
(November 1983).
11. S.fleinberg. Phya.Rev.Lett., 21» °57 (1976).
12. А.А.Ансельм, Н.Г.Уральцев. Я Ф , 30, 465 (1979).
13. S.Weinberg. Phys,Rev.Lett., 40, 223 (1978^.
14. F.ffilczek. Phys.Rev.Lett. 4^, ?7? (1975).
15. А.Р.ХитнишшЙ. ЯФ, 31, i97 (1980).
16. M.Dine, P.Piaher, M.Srednicki. Phys.Lett. 104B. 199 (1981).
17. M.B.Wise, H.Georgi, S.Glashow. Phys.Rev.Lett., £1, 402
(1981).
13. M.A.Shlfman, A.I.Vainshtein, V.I.Zakharov, Nucl.Phys.,
B166. 493 (1980).
19. A.A.Anselm, N.G.Uraltsev. Phys .Lett., 114B. 39; П С - . 161
(1982).
20. А.А.Ансельм. Письма в 1ЭТФ, 36, 46 (1982).
2 1 . G.B.Gelmini, S. Nussinov, T.Yanagida. Nucl.Phys., ,
(1983).
22. А.А.Аясельм, Н.Г.Уральцев. ЖЭТФ, 84, 1961 (1983).
23. P.Wilczeck. Phys.Rev.Lett., £J, 1549 (1982).37
24. Y.Chikashige, R.JT.Mohapatra, R.D.Peccei, Phys.Lett., J18B,
265 (1981); Phys.Rev.Lett., 4J5, 1926 (1980).
25. G.B.Gelmini, M.Roncadelli. Phya.Lett., 9JB, 411 (1981).
26. M.E.Peekin. Nucl.Phys.. B175. 197 (1980).
27. J . P r e s k i l l . Nucl.Phys., B177. 21 (1980).
28. Е.Б.Александров, А.А.Аясвльм, Ю.В.Павлов, Р.М.Умарходжавв.
ЖЭТФ, 85, 1890 (1983).
29. СНАВМ Collaboration. P h y s . L e t t . , 121B. 429 (1983).
30. R.Barbieri, S.Ferrara. CERN TH 3547.
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1983.
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Supplement. Gauge Theories; from 1919 till 193в,
Reprinta and Excerpts of Selected Paper» by P.Pock. P«London.
0. Klein and H.Weyl
1. H.Weyl. Elne neue Erweiterung der Relativitatstheorie.
Annalen der Physik j>9_, 101-133 (1919) (excerpts: p.p. 101,
114-115).
2. O.Klein. Quantum Theorie und funfdimensionale Relatiritats-
theorie.
Zeitschrift fiir Fhysik 3J, 895-906 (1926) (excerpts: p.p.
895, 904-906).
3* V.Pock. Ober die inrarianten Form der Wellen- und der
Bewegungsgleichungen fur einen geladenen Maesenpunkt.
Zeitschrift fur Physik ;J9_, 226-232 (1926).
4* P.London. Quantenmechanische Deutung der Theorie топ Weyl.
Zeitschrift fur Physik 42, 375-389 (1927) (excerpts: p.p.
375-379, 388. 389).
5» H.Weyl. Electron und Gravitation. I.
Zeitschrift fur Physik j>6, 330-352 (1929) (excerpts:
p.p. 330-333, 348, 349).
6. O.Klein. On the Theory of charged Fields, in "New Theories
in Physics". Conference organized in collaboration with the
International Union of Physics and the Polish Intellectual
Co-operation Committee. Warsaw, May, 30th - June 3rd, 1938.39
1919. JS 10.
ANNALEN DER PHYSIK
VIERTE FOLGE. UKD 59.
1. Eine neue Erweiterung der Relativitätstheorie;
von H. Weyl.
Kap. I. Oflomatriaeh« Grundlag«.
Einleitung. Um den physikalischen Zustand der Welt an
einer Weltstelle durch Zahlen charakterisieren zu können, muß
1. die Umgebung dieser Stelle aof Koordinaten bezogen sein
und müssen 2. gevrisse Maßeinheiten festgelegt werden. Die
bisherige Einst einsehe Relativitätstheorie bezieht sich nur
auf den ersten Punkt, die Willkürlichkeit des Koordinaten-
systems; doch gilt es, eine ebenso prinzipielle Stellungnahme
zu dem zweiten Punkt, der Willkürlichkeit der Maßeinheiten,
zu gewinnen. Davon soll im folgenden die E^de sein.
Die Welt ist ein vierdimensionales Kontinuum und läßt
sich deshalb auf vier Koordinaten xQ x, xt s, beziehen. Der
Übergang zu einem anderen Koordinatensystem x, wird durch
stetige Transformationsformeln
(1) *« = ft (Vi***a) (*' = 0, 1, 2, S)
vermittelt. An sich ist unter den verschiedenen möglichen
Koordinatensystemen keines aasgezeichnet. Die Relativ-
koordinaten dXi eines zu dem Punkte P = (a:,) unendlich
benachbarten P'—(x{+dxt) sind die Komponenten der in-
finitesimalen Verschiebung P P' (eines „Linienelementes" in
P). Sie transformieren sich beim Übergang (1) zu einem anderen
Koordinatensystem xt linear:
(2) dz.-^€cjdxk;
k
aj sind die Werte der Ableitungen dft}dxk im Punkfee P.
In der gleichen Weise transformieren sich die Komponenten !'
irgendeines Vektors in P. Mit einem die Umgebung von P
bedeckenden Koordinatensystem ist ein „Achsenkreuz" in P
verknüpft, bestehend aus den „Einheitsvektoren" e< mit den
Komponenten d,°, d,1,114 H.Weff
An seine Stelle aber trat bei Berücksichtigung der Gravitation
der Gegensatz von elektromagnetischem Feld („Materie im
weiteren Sinne", "wie Einstein sagt) und Gravitationsfeld; er
zeigt sich am deutlichsten in der Zweiteilung der Hamilton-
sehen Funktion, welche der £insteinschen Theorie zugrunde
liegt.1) Auch dieser Zwiespalt wird durch unsere Theorie
überwunden. Der Integr^nd der Wirkungsgröße /SB dz muß
eine aus der Metrik entspringende skalare Dichte SB sein, und
die Naturgesetze sind zusammengefaßt in dem Hamilton«
sehen Prinzip: Für jede infinitesimale Änderung 6 der Welt-
metrik, die außerhalb eines endlichen Bereichs verschwindet,
ist die Änderung
öj$&dx=fö$&dx
der gesamten Wirkungsgröße = 0 (die Integrale erstrecken
* sich über die ganze Welt oder, was auf dasselbe hinauskommt,
über einen endlichen Bereich, außerhalb dessen die Variation ö
verschwindet). Die Wirkungsgröße ist in unserer Theorie not-
wendig eine reine Zahl; anders kann es ja auch nicht sein, wenn
ein Wirkungsquantum existieren soll. Von SB werden wir an-
nehmen, daß es ein Ausdruck 2. Ordnung ist, d. h. aufgebaut
ist einerseits aus den g{k und deren Ableitungen 1. und 2. Ord-
nung, andererseits aus denI
41
Eine neue Erveiterung der RelativtiäUthcom. 115
Integral, dessen Integrand nur noch eine lineare Kombination
von &*!>;-.
42
Quantentheorie und fünfdimonsionalcRelativitätstheorie.
VIJII O.skur klein in Kopi uli.i:;. it.
(l-iiiiv'i ^iiiKi'ii »in - * • Ajiril l'.i.ii.)
Auf ili-n folp'mkn Seiten iin'icliti' icli ,iuf ini'ii einfachen Ztis.'iiiimenb.'in;: hm-
weisen «wischen der von Ksiliixa 1 ) \«»r^fsrhln^encii Theorie für den Zusammen
hanj; /.wischen Klektnmiacnetismus uml «iravitatmn einerseits und 1I1T von
ilr H r ii ^1 >«' *) '""' Sr hnid inger ') .IH^I'^CIICIICII Methode zur IlLliiiHiiluiiy der
Quiiuti'npnibleiiiC andererseits. I>ie Theorie vuu Knlur.n ^'t'lit darauf hinaus, du
zehn Kinstcinschrii (ir:ivitations|Hitintialc y.. und die vier clektroiiiiigneti.vlieh
i'iilcnli.ile , der gewtSiiniiclie» lielatix i-
tiitstlieorie zu kommen, müssen wir gewisse spezielle Annahmen m.nhen.
KrsdiiK müssen vier der Koordinaten, NII^CII wir ./•', ' s , ' 8 , r*, slet.- diu
/rcWidinliihrii Zeitraum ihaiaklerisiereti. Zweitens diirlon die (iittücn
') Th. KitlUKa, Sit/uii^her. d. iterl. Akad. I'.CM, S. '.Hill.
a
) \.. d. Urojjlie., Ann. d. I*II>-K. (10) », ii, J'.IL,5. ThAs-s, Paris JSKM.
') K. Sehr.idingKr, Ann. d. I'hys. 7«, '.«Ol und lö'J, 19-J'i.
Ziil.chrih lur rbyik. Hd. XXXVII. 55,43
904 Oikar Ckm,
rieh die Wellen nach den Gesetzen der geometrischen Optik ausbreiten.
Es mag noch hervorgehoben werden, daß wegen (42) die Beziehungen
(44), (45) bei den Koordinatentransformationen (2) invariant bleiben.
Betrachten wir nun auch die Gleichung (24) in dem Falle, wo u
nicht so groß ist, daß wir nur die in o quadratischen Glieder zu berück-
sichtigen brauchen. Wir beschranken uns dabei auf den einfachen Fall
eines elektrostatischen Feldes. Dann haben wir in "kartesischen Koor-
dinaten :
ds* = dy* x dp* X dr» — r» dt*. I
Also ergibt sich:
H = - (j>* X j>J + j>») - ^ t x e Vpt)* x ÜL*. ,«. (47)
In der Gleichung (24) können wir nun die mit |' ' proportionalen
Größen vernachlässigen, denn die Dreiindizessvmbole sind in diesem
Falle nach (17) kleine mit der Gravitationskonstante x proportionale
Größen. Wir bekommen also1):
!(?[' 2rV d*V , / . . *M-\44
Quateataeetie «adftkafdimemtioaaleSeUtmtltrtkMrie. 905
den ans der H e i s e n b e r g sehen Quantentheorie berechneten Energie-
werten identisch sind. Wie man sieht, ist £ in dem Grenzfall der geo-
metrischen Optik gleich der aal gewöhnliche Weise definierten mecha-
nischen Energie. Die Freqnenzbedingung besagt, wie S c h r ö d i n g e r
hervorgehoben hat, nach (51), daß die zu dem System gehörenden Licht-
frequenzen den ans den verschiedenen Werten der Frequenz v gebildeten
Differenzen gleich sind.
§ 3. Schlußbemerkungen'. Wie die Arbeiten von de B r o g l i e
sind obenstehende Überlegungen ans dem Bestreben entstanden, die
.Analogie zwischen Mechanik und Optik, die in der Hamiltonschen
Methode zum Vorschein kommt, für ein tieferes Verständnis der Quanten-
erscheinungen auszunutzen. Daß dieser Analogie ein reellei physi-
kalischer Sinn zukommt, scheint ja die Ähnlichkeit der Bedingungen für
die stationären Zustände mn Atnmsystemen mit den Interferenz-
erscheinungen der Optik anzudeuten. Nun stehen bekanntlich Begriffe
wie Punktladung und materieller Punkt schon der klassischen Feld-
pbvsik fremd gegenüber. Auch wurde ja öfters die Hypothese aus-
gesprochen, daß die materiellen Teilchen als spezielle Lösungen der
Feldgleichungen aufzufassen 6ind, welche das Gravitationsfeld und da=
elektromagnetische Feld bestimmen. Es liegt nahe, die genannte Ana-
logie zu dieser Verstellung in Beziehung zu bringen. Denn nach dieser
Hypothese ist es ja nicht s>> befremdend, daß die Bewegung der mate-
riellen Teikben Ähnlichkeiten aufweist mit der Ausbreitung V>JU Well^i,
l.He in Hede stehende Analogie ist jedoch unvollständig, solange man
eine Wellenausbreitung in einem Raum von nur vier Dimensionen !*•-
trachtet. Dies kommt schon in der variablen Gesehwindiffkeit der
materiellen Teilchen zum Vorschein. Deukt man sich aber die beob-
achtete Bewegung als eine Art Projektion auf den Zeiträum von einer
YVcileuausbieitung. die in einem Kaum von fünf Dimensionen stattfindet,
%h läßt sich, wie wir sahen, die Analogie vollständig machen. Mathe-
matisch ausgedrückt heißt dies, daß die Hamilton-.Tacohische Glei-
chung nicht als CharakteristikeugleiHiung einer vierdiroensionalen. wohl
aber einer fünfdimensi'malen Wellcngleiehung aufgefal't wor.len Linn
lu dieser Weise wird man zu der Theorie von K a l u z a geführt.
Oliwohl die Einführung einer fünften Dimension in u«*ere physi-
kalischen Betrachtungen von vornherein befremdend sein mag. wird eine
radikale Modifikation der den Feldgleichungen zugrunde selegten
Geometrie doch wieder in ganz anderer Weise durch die Quantentheorie
nahegelegt Denn es ist bekanntlich immer weniger wahrscheinlich45
90ft Oskar Klein, Quantentheorie und tamfdiiiienaioaale Relativitätstheorie.
geworden, daß die Cjnantenerscheinnngen eine einheitliche raamzeitliche
Beschreibung zulassen, wogegen die Möglichkeit, diese Erscheinungen
durch ein System von fünfdimensionalen Feldgleichungen darzustellen,
wohl nicht von vornherein auszuschließen ist l). Ob hinter diesen An-
deutungen von Möglichkeiten etwas Wirkliches besteht, muß natürlich
die Zukunft entscheiden. Jedenfalls muß betont werden, daß die in dieser
Note versuchte Behandlungsweise. sowohl \va< die Feldgleichungen als
auch die Theorie der stationären Zustände betrifft, al> canz provisorisch
zu betrachten ist. Dies kommt wohl besonders in der auf >. ,-r*> er-
wähnten schematischen Behandlungsweise der Materie zum Vorschein,
•=o\vic in dem Umstand, daß die zwei Arten von elekrris< heu Teilchen
durch verschiedene Gleichungen vom S c h r ö d i n ^ e r s e h e n Typu- behandelt
werden. Auch wird die Frage ganz offen gelassen, ob man siih bei der
Beschreibung der physikalischen Vorsänge mit den 14 Potentialen be-
gnügen kann, oder ob die S c h r ö d i n g e r s e h e Methode dit Einführung
einer neuen Zustandsgröße bedeutet.
Mit den in dieser Xote mitgeteilten L'berleiruniren iiai»- ich mich
sowohl in ' dem Physikalischen Institut der Fniver-itv of Michisran. ABU
Arbor. wie in ilent hiesigen Institut für theoretische Physik beschäftigt.
Ich möchte auch aii dieser Stelle Prof. H. M. H a n d a l i und Prot. N" Bohr
meinen wärmsten Dank aussprechen.
'i Bemerkungen diese Art. dl* Prof. Bohr l»-i iite'irt-r-n «r^lt-cenivit» u
gfmaWn iiat. li.iii-.-a ein«-n enisrhicj-neu Eioiiull auf d;;- Enwrhrn der \'»r-
li'^en'len Xot- schabt.You can also read
