Quantum mechanics and quantum field theory

28-2 On the relation between the integrals of the quantummechanical equations of motion and the Schr¨odinger 29-3 On the Dirac equations in general relativity 10929-4 Dirac wave equation

Quantum Mechanics and Quantum Field Theory

V.A Fock

CHAPMAN & HALL/CRC

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Printed on acid-free paper

Library of Congress Cataloging-in-Publication Data

Fock, V A (Vladimir Aleksandrovich), 1898-1974 [Selections English 2004]

V.A Fock selected works : quantum mechanics and quantum field theory / by L.D Faddeev, L.A Khalfin, I.V Komarov.

p cm.

Includes bibliographical references and index.

ISBN 0-415-30002-9 (alk paper)

1 Quantum theory 2 Quantum field theory I Title: Quantum mechanics and quantum field theory II Faddeev, L D III Khalfin, L A IV Komarov, I V V Title.

QC173.97.F65 2004

Visit the CRC Press Web site at www.crcpress.com

28-2 On the relation between the integrals of the quantum

mechanical equations of motion and the Schr¨odinger

29-3 On the Dirac equations in general relativity 10929-4 Dirac wave equation and Riemann geometry 11330-1 A comment on the virial relation 13330-2 An approximate method for solving

the quantum many-body problem 13730-3 Application of the generalized Hartree method

30-4 New uncertainty properties of the electromagnetic field 177

32-1 A comment on the virial relation in classical mechanics 18732-2∗ Configuration space and second quantization 19132-3∗ On Dirac’s quantum electrodynamics 22132-4∗ On quantization of electro-magnetic waves and

interaction of charges in Dirac theory 22532-5∗ On quantum electrodynamics 24333-1∗ On the theory of positrons 257

vi CONTENTS

34-1 On the numerical solution of generalized equations of

34-2 An approximate representation of the wave functions

34-3∗ On quantum electrodynamics 33135-1 Hydrogen atom and non-Euclidean geometry 36935-2 Extremal problems in quantum theory 38136-1 The fundamental significance of approximate methods

37-1∗ The method of functionals in quantum electrodynamics 40337-2∗ Proper time in classical and quantum mechanics 42140-1 Incomplete separation of variables for divalent atoms 44140-2 On the wave functions of many-electron systems 46743-1 On the representation of an arbitrary function by an

integral involving Legendre’s function with a complex

On December 22, 1998 we celebrated the centenary of Vladimir sandrovich Fock, one of the greatest theoretical physicists of the XX-thcentury V.A Fock (22.12.1898–27.12.1974) was born in St Petersburg.His father A.A Fock was a silviculture researcher and later became aninspector of forests of the South of Russia During all his life V.A Fockwas strongly connected with St Petersburg This was a dramatic period

Alek-of Russian history — World War I, revolution, civil war, totalitarianregime, World War II He suffered many calamities shared with the na-tion He served as an artillery officer on the fronts of World War I, passedthrough the extreme difficulties of devastation after the war and revo-lution and did not escape (fortunately, short) arrests during the 1930s.V.A Fock was not afraid to advocate for his illegally arrested colleaguesand actively confronted the ideological attacks on physics at the Soviettime

In 1916 V.A Fock finished the real school and entered the department

of physics and mathematics of the Petrograd University, but soon joinedthe army as a volunteer and after a snap artillery course was sent tothe front In 1918 after demobilization he resumed his studies at theUniversity

In 1919 a new State Optical Institute was organized in Petrograd, andits founder Professor D.S Rozhdestvensky formed a group of talentedstudents A special support was awarded to help them overcome thedifficulties caused by the revolution and civil war V.A Fock belonged

to this famous student group

Upon graduation from the University V.A Fock was already the thor of two scientific publications — one on old quantum mechanics andthe other on mathematical physics Fock’s talent was noticed by theteachers and he was kept at the University to prepare for professorship.From now on his scientific and teaching activity was mostly connectedwith the University He also collaborated with State Optical Institute,Physico-Mathematical Institute of the Academy of Sciences (later splitinto the Lebedev Physical Institute and the Steklov Mathematical In-stitute), Physico-Technical Institute of the Academy of Sciences (laterthe Ioffe Institute), Institute of Physical Problems of the Academy ofSciences and some other scientific institutes

au-Fock started to work on quantum theory in the spring of 1926 just ter the appearance of the first two Schr¨odinger’s papers and in that same

af-viii Preface

attracted attention and in 1927 he received the Rockefeller grant for oneyear’s work in G¨ottingen and Paris His scientific results of this periodorists of the world The outstanding scientific achievements of V.A Fockled to his election to the USSR Academy of Sciences as a correspondingmember in 1932 and as an academician in 1939 He was awarded thehighest scientific domestic prizes The works by V.A Fock on a widerange of problems in theoretical physics — quantum mechanics, quan-tum field theory, general relativity and mathematical physics (especiallythe diffraction theory), etc — deeply influenced the modern develop-ment of theoretical and mathematical physics They received worldwiderecognition Sometimes his views differed from the conventional ones.Thus, he argued with deep physical reasons for the term “theory ofgravitation” instead of “general relativity.” Many results and methodsdeveloped by him now carry his name, among them such fundamentalones as the Fock space, the Fock method in the second quantization the-ories, the Fock proper time method, the Hartree–Fock method, the Focksymmetry of the hydrogen atom, etc In his works on theoretical physicsnot only had he skillfully applied the advanced analytical and algebraicmethods but systematically created new mathematical tools when theexisting approaches were not sufficient His studies emphasized the fun-damental significance of modern mathematical methods for theoreticalphysics, a fact that became especially important in our time

In this volume the basic works by Fock on quantum mechanics andquantum field theory are published in English for the first time Aconsiderable part of them (including those written in co-authorship with

M Born, P.A.M Dirac, P Jordan, G Krutkov, N Krylov, M Petrashen,

B Podolsky, M Veselov) appeared originally in Russian, German orFrench A wide range of problems and a variety of profound resultsobtained by V.A Fock and published in this volume can hardly be listed

in these introductory notes A special study would be needed for the fulldescription of his work and a short preface cannot substitute for it.Thus without going into the detailed characteristics we shall specifyonly some cycles of his investigations and some separate papers Webelieve that the reader will be delighted with the logic and clarity ofthe original works by Fock, just as the editors were while preparing thisedition

In his first papers on quantum mechanics [26-1, 2] Fock introducesthe concept of gauge invariance for the electromagnetic field, which hecalled “gradient invariance,” and, he also presents the relativistic gener-year he published his own two papers on this subject (see [26-1,2]) They

(see [28-1,2,3,4]) placed him at once in the rank of the most active

the-Preface ix

alization of the Shr¨odinger equation (the Klein–Fock–Gordon equation)that he obtained independently and simultaneously with O Klein andearlier than W Gordon In a series of works [29-3, 4] on the geometriza-tion of the Dirac equation Fock gives the uniform geometrical formula-tion of gravitational and electromagnetic fields in terms of the generalconnection defined not only on the space–time, but also on the internalspace (in modern terms) In the most direct way these results are con-nected with modern investigations on Yang–Mills fields and unification

of interactions

Many of Fock’s works [30-2, 3; 33-2; 34-1, 2; 40-1, 2] are devoted toapproximation methods for many-body systems based on the coherenttreatment of the permutational symmetry, i.e., the Pauli principle Let

us specifically mention the pioneer publication [35-1], where Fock wasthe first to explain the accidental degeneracy in the hydrogen atom bythe symmetry group of rotations in 4-space Since then the dynamicalsymmetry approach was extensively developed In the work [47-1] animportant statement of the quantum theory of decay (the Fock–Krylovtheorem) was formulated and proved, which has become a cornerstonefor all the later studies on quantum theory of unstable elementary (fun-damental) particles

A large series of his works is devoted to quantum field theory [32-1,

2, 3, 4, 5; 34-3; 37-1] In those works, Fock establishes the coherenttheory of second quantization introducing the Fock space, puts forwardthe Fock method of functionals, introduces the multi-time formalism

of Dirac–Fock–Podolsky etc The results of these fundamental worksnot only allowed one to solve a number of important problems in quan-tum electrodynamics and anticipated the approximation methods likethe Tamm–Dankov method, but also formed the basis for subsequentworks on quantum field theory including the super multi-time approach

of Tomonaga–Schwinger related to ideas of renormalizations It is ticularly necessary to emphasize the fundamental work [37-2] in whichFock introduced an original method of proper time leading to a newapproach to the Dirac equation for the electron in the external electro-magnetic field This method played an essential role in J Schwinger’sstudy of Green’s functions in modern quantum electrodynamics.The new space of states, now called the Fock space, had an extraordi-nary fate Being originally introduced for the sake of consistent analysis

par-of the second quantization method, it started a new independent life inmodern mathematics The Fock space became a basic tool for studyingstochastic processes, various problems of functional analysis, as well as

in the representation theory of infinite dimensional algebras and groups

x Preface

Besides theoretical physics, Fock also worked in pure mathematics.For this edition we have chosen two such works most closely related toquantum physics In [29-1], published only one year after delta-functionswere introduced by Dirac, Fock obtained the rigorous mathematicalbackground for these objects unusual for classical analysis, and thuspreceded the further development of the theory of generalized functions

In [43-1] Fock gave an original representation of an arbitrary function by

an integral involving Legendre’s functions with a complex index Later,this work entered the mathematical background of the well-known Reggemethod

As time goes on, the significance of works of classics of science — andFock is undoubtedly such a classic — becomes more and more obvious.The fame of brilliant researchers of new particular effects, sometimesrecognized by contemporaries higher than that of classics, is perhapsless lasting This is not surprising, for in lapse of time more simple andmore general methods appear to deal with particular effects, while theclassical works lay in the very basis of the existing paradigms Certainly,when a paradigm changes (which happens not so often), new classicsappear However, it does not belittle the greatness of the classics as towho founded the previous paradigm So the discovery of quantum theory

by no means diminished the greatness of the founders of classical physics

If in the future the quantum theory is substituted for a new one, it by

no means will diminish the greatness of its founders, and in particularthat of Fock His name will stay forever in the history of science

As a real classic of science Fock was also interested in the ical concepts of new physics In this volume we restricted ourselves toonly two of his papers on the subject [47-1, 57-1] Fock fought againstthe illiterate attacks of marxist ideologists on quantum mechanics andrelativity His philosophical activity helped to avoid in physics a pogrom

philosoph-of the kind suffered by Soviet biology

The works by Fock were translated into English and prepared for thisedition by A.K Belyaev, A.A Bolokhov, Yu.N Demkov, Yu.Yu Dmitri-

ev, V.V Fock, A.G Izergin, V.D Lyakhovsky, Yu.V Novozhilov, Yu.M.Pis’mak, A.G Pronko, E.D Trifonov, A.V Tulub, and V.V Vechernin.Most of them knew V.A Fock, worked with him and were affected byhis outstanding personality They render homage to the memory of theirgreat teacher

Often together with the Russion version Fock published its variant inone of the European languages, mainly in German We give references

to all variants Papers are in chronological order and are enumerated

by double numbers The first number indicates the year of first

publi-Preface xi

cation To distinguish papers published the same year we enumeratedpaper in this issue originally published in 1934 In 1957 the collectedpapers by Fock on quantum field theory were published by Leningrad

University Press in V.A Fock, Raboty po Kvantovoi Teorii Polya, datel’stvo Leningradskogo Universiteta, 1957 The articles for the book

Iz-were revised by the author In the present edition papers taken fromthat collection are shown with an asterisk

The editors are grateful to A.G Pronko who has taken on the burden

of the LATEX processing of the volume

We believe that the publication of classical works by V.A Fock will

be of interest for those who study theoretical physics and its history

L.D Faddeev, L.A Khalfin, and I.V Komarov

St Petersburg

them by the second number Hence the reference [34-2] means the second

JRPKhO — Journal Russkogo Fiziko-Khemicheskogo

Obshchestva, chast’ fizicheskaja

JETP — Journal Eksperimentalnoi i Teoreticheskoi Fiziki

DAN — Doklady Akademii Nauk SSSR

Izv AN — Izvestija Akademii Nauk SSSR,

serija fizicheskaja

UFN — Uspekhi Fizicheskikh Nauk

Vestnik LGU — Vestnik Leningradskogo Gosudarstvennogo Universiteta,

serija fizicheskaja

UZ LGU — Ucheniye Zapiski Leningradskogo Gosudarstvennogo

Universiteta, serija fizicheskikh nauk

OS — Optika i Spektroskopija

Fock57 — V.A Fock, Raboty po Kvantovoi Teorii Polya Izdatel’stvo

Leningradskogo Universiteta, Leningrad, 1957

On Rayleigh’s Pendulum

G Krutkov and V Fock

Petrograd Received 12 December 1922

Zs Phys 13, 195, 1923

The importance of Ehrenfest’s “Adiabatic Hypothesis” for the presentand future of the quantum theory makes very desirable an exact exam-ination of its purely mechanical meaning Some years ago one of theauthors1 found a general method to look for the adiabatic invariants,whereas the other author2investigated the case of a degenerated, condi-tionally periodic system which had not been considered in the first paper

3

to this theory is that in the course of calculations at some point a plifying assumption was made in the integrated differential equations,namely that its right-hand side is subject to an averaging process; toexplain this approximation, arguments connected with the slowness ofchanges of the system parameters were used This shortcoming makes

sim-it difficult to use the ordinary methods and to check the adiabatic variance of several quantities Therefore it seems reasonable to consider

in-a very simple exin-ample which we cin-an integrin-ate without in-any in-additionin-alassumptions and only then use the slowness of parameter changes

As such an example we chose the Rayleigh pendulum, which is sic” for the “adiabatic hypothesis,” i.e., a pendulum the length of which

“clas-is changing continuously but the equilibrium point remains fixed As “clas-iswell known the adiabatic invariant here is the quantity

v = E

ν ,

the relation of the energy of the pendulum to the frequency This

1 G Krutkow, Verslag Akad Amsterdam 27, 908, 1918 = Proc Amsterdam 21, 1112; Verslag 29, 693, 1920 = Proc 23, 826; TOI 2, N12, 1–89, 1921.

2 V Fock, TOI 3, N16, 1–20, 1923.

3 J.M Burgers, Ann Phys 52, 195, 1917.

(see also the paper by Burgers ) An objection which can be attributed

2 G Krutkov and V Fock

quantity can be also written in the form of the action integral

aver-2 and 3 The general proofs of the adiabatic invariance of the “phase

integral” and the above mentioned general theory prove the v-invariance

as a special case.5,6Here in the course of calculations we neglected someterms, too; and finally

4 The variational principle

23-1 On Rayleigh’s pendulum 3

1 The General Method Establishing

the Differential Equations of the Problem

A system has f degrees of freedom Its Hamilton function8 depends on

the (generalized) coordinates q r , momenta p r , and on parameter a

we put a = const and integrate the resulting isoparametric problem, then

we obtain f integrals of motion

H1= c1 , H2= c2 , , H f = c f (c) which are in involution Then we solve them relative to p r

p r = K r (q1, , q f , c1, , c f ; a) , (c 0)and form the Jacobi characteristic function

tion V (q1, , q f , c1, , c f ; a) Now we remove the condition a = const;

a can be an arbitrary function of time t We come then to the metric problem According to a known theorem the differential equations

rheopara-for the “elements” remain canonical with the new Hamilton function

8 V.A Fock avoided the use of the currently common word “Hamiltonian” saying

that it sounded to him like an Armenian name (Editors)

4 G Krutkov and V Fock

where the brackets mean that the derivative of V should be expressed through c r , ϑ r Thus the “rheoparametric equations” are

The next step, the averaging process, should not be performed

Now we turn to the Rayleigh pendulum We make a preliminarycondition that we stay in the region of small oscillations which is a re-striction for the change of the pendulum length; actually by a largeenough shortening of the length we shall come to the non-small elonga-tion angles However this restriction is not essential because we shallfurther assume that the velocity of the length decrease is small For the

pendulum length λ we put

with α = const, i.e., we consider the case of a constant velocity of the parameter change.

If the mass of a heavy point is equal to 1, ϕ is the angle of elongation,

p = λ2ϕ is the angular momentum and g is the gravity acceleration, then˙

we have the Hamilton function

H = 12λ2 p2+1

2 g λ ϕ

2. (2)

If we put H = c, we have

p =p2λ2c − gλ3ϕ2 (20)and

α (5)The rheoparametric equations for our problem are

dc

dt = α

µ

c 2λ+

3 c

2 λ cos ϑ 2

r

g λ

where for λ one must substitute λ = l − α t.

2 Integration of the Differential Equations (∗)

Because equation (B) does not contain the variable c it can be considered

separately We put

x = 2

√ g α

dx ,

6 G Krutkov and V Fock

after that the differential equation will have the form:

d2u

dx2 −3x

Before we go further we shall check the results obtained by establishing

the equation of motion for the deviation angle ϕ of the pendulum and

by integration of this equation

9 See, e.g., P Schafheitlin, Die Theorie der Besselschen Funktionen, p 123,

for-mulas 4(1) and 4(2) (Authors)

23-1 On Rayleigh’s pendulum 7

Fig 1

For the coordinates of the point 1 which remains in the (x, y)-plane and keeps the distance l − η from point 2 (see Fig 1) which in turn lies on the y axis at a distance η from the origin, we have

x = (l − η) sin ϕ , y = (l − η) cos ϕ + η and consequently for the kinetic energy T and potential energy Π

8 G Krutkov and V Fock

√

λ = 2σ

dy

dx + y = 0

The solution, as one easily finds, is:

y = x[A J1(x) + B Y1(x)] = x Z1(x) For the angle ϕ it follows:

the sum of which by definition is the quantity c (see (2)) which we can

consider as the energy of the pendulum:

c = 2gl

σ2 [Z2(x) + Z2(x)] , (18)

in complete accordance with formula (9)

¶

− ϕ0 J2

µ2

To prove the adiabatic invariance relative to v, we assume that the length

of the pendulum decreases slowly, i.e., α is small, whereas σ and x are large; the latter assumption demands στ ¿ 1 We can now replace in

Z k (x) (k = 1, 2) the J k (x) and Y k (x) and J k

µ2

σ

¶

, Y k

µ2

σ

¶, which enter

formulas (19) for A and B, by their asymptotic expressions:

J k (x) =

r2

σ − x

¶

− ϕ0sin

µ2

σ − x

¶

+ ϕ0cos

µ2

and for c:

c = 2 g l

σ2 · σ 2x

10 G Krutkov and V Fock

In this paper we will try to remove some of these difficulties and tofind the corresponding wave equation for the more general case when theLagrange function contains the linear (in velocities) terms.

Our paper consists of two parts

method Schr¨odinger has already obtained some of these results, butonly the results themselves; the calculations were not submitted

In part I the wave equation isformulated; part II contains examples of the Schr¨odinger quantization

12 V Fock

The left-hand side of this equation is a quadratic function of the

deriva-tives of the function of action W with respect to coordinates.1

we have the uniform quadratic function of the first derivatives

of ψ relative to the coordinates and time:

The integration over the coordinates should be extended over the

whole coordinate spaces and over an arbitrary time interval t2> t > t1

1This is in classical mechanics The relativistic mechanics of a single point-like

mass (at least without a magnetic field) allows the equation to be written in this form; however it looks like the operations connected with the transformed equation

are not completely inarguable (V Fock)

26-1 On Schr¨ odinger’s wave mechanics 13

One can obtain the required wave equation by putting the first

vari-ation of the integral (functional) J equal to zero:

If one looks for periodic solutions and puts

ψ = e 2πiνt ψ1= e i E

one obtains for ψ1an equation which does not contain time The energy

E enters here as a parameter and at least in the nonrelativistic ics E enters linearly In the special case of vanishing P i equation (3)

mechan-coincides with that established by Schr¨odinger If P i does not vanish,the coefficients of the (time-independent) wave equation are complex.The special energy values can be found using the uniqueness, finite-ness and continuity conditions of the solutions

For pure periodic solutions (6) one can put directly the expressions

1 The Kepler Motion in Magnetic Field

Let us consider a uniform magnetic field of strength H oriented along the z-axis The Lagrange function

are well known.

The quadratic form Q is

Q = E

2

2m (grad ψ)

2− eHE 2mc

∆ψ − eH Ec

m

~ ωa

2= ω1 , 2Ea

26-1 On Schr¨ odinger’s wave mechanics 15

If we neglect ω2(weak magnetic field), the equation can be solved if we

look for ψ1in the form (separation in spherical coordinates)

ψ1= e in1ϕ P n1

n (cos ϑ) r n ψ2(r) , (16)

where P n1

n (cos ϑ) are the “adjoint spherical functions.” Then for ψ2(r)

we have the equation

which agrees with the old quantum theory

2 The Motion of an Electron in the Electrostatic Field of theNucleus and in the Magnetic Field of the Dipole whichCoincides with the Nucleus2

Solving this problem we meet the difficulty of a general nature, which

we cannot overcome here The example is chosen just to draw attention

to the possibility of these difficulties

Let the z-axis coincide with the direction of the dipole momentum of the value M The Lagrange function is

16 V Fock

we see that the point r = 0 is an essentially singular point (the

uncer-tainty position) for all integrals To understand this more clearly, let us

choose quantity a (12) as a unit of length

r3 ; from the physical point of view this term must play the role

of a small correction3 and by no means could be essentially importantfor the solution This difficulty is characteristic not only of the chosenexample, but occurs in all cases where we consider the approximatepresentation of forces; in the theory of the Schr¨odinger wave equation wehave to consider these forces not only in the region of electronic orbits,but in the whole space In a “natural” mechanical system (electronsand nuclei) this difficulty would possibly arise It is now not clear how

to overcome it Probably we have to use different approximations offorces in different regions of space and put some continuity conditions

for the wave function ψ at the borders of these regions Whether any

ambiguity in the evaluation of the eigenenergies could then be excludedremains unclear Anyway, the question touched upon here needs a deeperinvestigation

3 The Relativistic Kepler Motion4

The (HJ) equation (cleared from the square roots) has the form

3We have already neglected the squares of β (V Fock)

4 See footnote 2 (V Fock)

26-1 On Schr¨ odinger’s wave mechanics 17

and the corresponding wave equation is

We introduce the quantity a1as a unit of length and make the periodicity

assumption (6) Then the equation for ψ will be

18 V Fock

4 The Stark Effect

Again we orient the direction of the electric uniform field with the

strength D along the z-axis We introduce the parabolic coordinates

z + iρ = a

2(ξ + iη)

2, (35)use the notations

We introduce new variables x, y and new parameters λ(1) , λ(2) , µ

ξ2= √2

−α x , η

2=√2

−α y , 2² = µ( √ −α)3, √1

26-1 On Schr¨ odinger’s wave mechanics 19

Both equations have the form

Now we use the Laplace transformation

Here µ is a small parameter of the order of the electric field strength.

We look now for the expansions of λ, F (t), f (z) in powers of µ:

be a rational function, therefore the function F0 must be an

integer-transcendent one, so that in the first equation (41) λ0 must be equalto

λ(1)0 = n − 1 + 2p1 (p1= 1, 2, ) , (48a)

and in the second equation (41)

λ(2)0 = −n + 1 − 2p2 (p2= 1, 2, ) (48b)

1 E Schr¨odinger, Quantisiering als Eigenwertproblem, Ann Phys 79,

361 (I Mitteilung) and 79, 489 (II Mitteilung), 1926.

2 G Krutkow, Adiabatic Invariants and their Application in Theoretical Physics, TOI 2, N12, 38, 1922 (in Russian).

Leningrad,

Physical Institute of the University

Translated by Yu.N Demkov

Zs Phys 39, N2–3, 226–232, 1926

In his not yet published paper H Mandel2 uses the notion of the dimensional space for considering the gravity and the electromagneticfield from a single point of view The introduction of the fifth coor-dinate parameter seems to us very suitable for representing both theSchr¨odinger wave equation and the mechanical equations in the invari-ant form

five-1 The special relativity

The Lagrange function for the motion of a charged massive point is, ineasily understandable notations,

1 The idea of this work appeared during a discussion with Prof V Fr´eederickcz,

to whom I am also obliged for some valuable pieces of advice (V Fock)

Remark at proof When this note was in print, the excellent paper by Oskar Klein

(Zs Phys 37, 895, 1926) was received in Leningrad where the author obtained the results which are identical in principle with the results of this note Due to the importance of the results, however, their derivation in another way (a generalization

of the Ansatz used in my earlier paper) may be of interest (V Fock)

For history of the subject see Helge Kragh Equation with many fathers

Klein-Gordon equation in 1926 Am J Phys 52, N 11, 1024–1033, 1984 (Editors)

2 The author kindly gave me a possibility to read his paper in manuscript.

(V Fock)

2

c2(U2− ϕ2) = 0. (2)Analogously to the Ansatz used in our earlier paper3 we put here

∂p are four-dimensional invariants Further, the form Q remains

invariant if one puts

where f denotes an arbitrary function of the coordinates and of time.

The latter transformation also leaves invariant the linear differentialform4

3V Fock, Zur Schr¨ odingerschen Wellenmechanik, Zs Phys 38, 242, 1926 (see

4The symbol d 0 means that d 0 Ω is not a complete differential (V Fock)

[ 26-1 ] in this book).

26-2 On the invariant form of 23

We will now represent the form Q as the squared gradient of the tion ψ in the five-dimensional space (R5) and look for the correspondingspace interval One can easily find

µ

U grad ∂ψ

∂p +

ϕ c

∂ψ

∂p

µdiv U +1

c

∂ϕ

∂t

¶+

ordinate parameter p is exactly that it ensures the invariance of the

equations with respect to the addition of an arbitrary gradient to thefour-potential

It is to be noted here that the coefficients of the equation for ψ0arecomplex-valued in general

Assuming further that these coefficients do not depend on t and

Both functions ψ1 and ψ1 corresponding to the vector potentials U and

5The appearance of the parameter p, connected with the linear form, in the

ex-ponent can be possibly associated with some relations observed by E Schr¨ odinger

(Zs Phys 12, 13, 1923) (V Fock)

24 V Fock

U = U − grad f differ, namely, by a factor e 2πie ch f of modulus 1, thus only

possessing (at very general assumptions for the function f ) the same

continuity properties

2 The General Relativity

A The wave equation For the space interval in the five-dimensionalspace we put

The quantities g ik here are the components of the Einstein

funda-mental tensor, the quantities q i (i = 1, 2, 3, 4) being the components of the four-potential divided by c2, so that

The quantity q5 is a constant, and x5 is the additional coordinate

pa-rameter All the coefficients are real-valued being independent of x5

The quantities g ik and q i depend on the fields only but not on theproperties of the massive point, the latter being represented by the factor

introducing also the following agreement: at the summation from 1 to

5, the sign of the sum is written explicitly, and at the summation from

1 to 4 it is suppressed

Using these notations we find

γ ik = g ik + a i a k; g i5 = 0, (i, k = 1, 2, 3, 4, 5), (14)

γ =k γ ik k= a25g, (i, k = 1, 2, 3, 4, 5), (15)

26-2 On the invariant form of 25

26 V Fock

Then we have

nkl r

o

5 =nkl r

o

4+1

2(a k g

ir M il + a l g ir M ik ), nkl

d 0 Ω = a i dx i + a5dx5. (25)

Multiplying the four equations (23) by a r , the fifth equation (24) by a5

and summing up, one obtains an equation which can be written in theform

d ds

¶

which, after introducing the proper time τ by the formula

g ik dx i dx k = −c2dτ2, (29)

26-2 On the invariant form of 27

gives

d ds

µ

dτ ds

To get the Hamilton–Jacobi equation, we put the square of the

five-dimensional gradient of function ψ equal to zero,