Born–Infeld Theory

5 Born–Infeld Theory

In 1934, M. Born and L. Infeld published a paper on “The Foundations of the New Field Theory” [42 *]. Its somewhat vague title hid a non-linear theory of the electromagnetic field using a non-symmetric metric but denying a relationship with “ ‘unitary‘ field due to Einstein, Weyl, Eddington, and others […]”. In fact, the original idea for the new theory originated in July 1933 while Born was still a member of the University of Göttingen but already on the move from Germany for vacations in South Tyrol to only return after World War II. Born’s next publication, submitted in August 1933 without institutional address, dealt with the quantization of the electromagnetic field; in it the new Lagrangian was also shown [37 *]. In view of the problems with divergent terms in quantum (field) electrodynamics at the time, he set out to modify Maxwell’s equations in such a way that an electron with finite radius $r0$ could be described; its electric potential remained finite for $rr0 ≪ 1$ [36 ]. The Lagrangian for the new electrodynamics was $----------------- L = -1∘ 1 + a2(H2 − E2 ) a2$ with the constant a of dimension $r2 e0$ , where $e$ is the elementary electric charge and $r0$ the electron radius. In the limit $a → 0$ the Lagrangian of Maxwell’s theory reappeared: $1a2 + 12(H2 − E2)$ . In the paper with Infeld, the Lagrangian is generalized in order to include the gravitational field:

where $g(ij)$ is the (Riemannian) metric and $fij = f[ij]$ the electromagnetic field tensor; $gij = g(ij) + fij$ , formally is an asymmetric metric. The Lagrangian (147 *) can be expressed by the two invariants of Maxwell’s theory $F := 1fmrfnsgmngrs 2$ and $G := 1𝜖mnrsfmnfrs == f∗rsfrs 4$ as

The new field equations become:

with the definition $√ −-g pik := -∂ℒ- ∂fik$ . Insertion of $→ → → → F = 1∕b2(B2 − E2 ), G = 1∕b2B. E$ led to Maxwell’s equations plus the relations between fields and inductions:

“The quotient of the field strength expressed in the conventional units divided by the field strength in the natural units” was denoted by $b$ and named the “absolute field”. As was well known, many asymmetric energy-momentum tensors for the electromagnetic field could be formulated. Years later, St. Mavridès took up this problem and derived identities for the symmetric Minkowski tensor, the fields and inductions, independent of whether the relations between fields and inductions were linear or more general [410 ]. Xinh Nguyen Xua then showed that with the relations (150 *), all the various energy-momentum tensors can be derived from one such symmetric tensor [711 ]. Born & Infeld chose

The static solution of the new equations for the potential of a point charge was determined to be

It turned out that, from $2 b2 ∫ 3 E = m0c := 4π Ld x$ , $r0$ could be calculated numerically via $∫ 2 e2r( 0∞(1 − √-x2-4) 0 1+x$ to take the value $r0 = 1,2361m-e2c2 0$ and thus $b$ could also be determined. According to Born and Infeld: “The new field theory can be considered as a revival of the old idea of the electromagnetic origin of mass” Also, the existence of an absolute field as a “natural unit for all field components and the upper limit for a purely electric field” ([42 *], p. 451) had been assumed.

Unsurprisingly, Pauli was unhappy with the paper by Born and Infeld as far as its inclusion of the gravitational field via $g + f (ij) ij$ was concerned. Instead, in his letter of 21 December 1933 to Max Born ([488 *], p. 241), he suggested to take as a Lagrangian density $√ --- ℒ= − gF (P)$ with Kaluza’s curvature scalar in 5 dimensions $P = R + -k2frsf rs 2c$ . “In particular, it is possible to set $∘ ----------- F(P ) = 1 + --1--P const.$ , and therefore your electrodynamics is compatible with the projective view on the electromagnetic and gravitational field.”⁶⁸ But for Born, electrodynamics was in the focus. Three months after Pauli’s criticism, he wrote to Einstein in connection with his paper with Infeld:

“Possibly, you will not agree, because I do not include gravitation. This is a rather basic point, where I have a different view as you in your papers on unitary field theory. Hopefully, I soon will be able to finalize my idea on gravitation” ([168 *], p. 167).⁶⁹

Around the same time as Pauli, B. Hoffmann who had left Princeton for the University of Rochester, had had the same idea. It was couched in the language of projective theory on which he had worked with O. Veblen (cf. Section 6.3.2 of Part I) and on his own [275 *].⁷⁰ He suggested the Lagrangian $√ ------ √ ---- ℒ= ( 1 + B − 1 ) − γ$ where $γij$ is the 5-dimensional projective metric and $B$ the projective curvature scalar. Due to $B = R − gprgqsfrqfps$ , his Lagrangian corresponds to Kaluza’s. Born & Infeld had remarked that in order to include gravitation in their theory, only Einstein’s Lagrangian must be added to (148 *). Hoffmann now tried to obtain a static spherically symmetric solution for both theories with a non-vanishing electromagnetic field. In the augmented Born–Infeld Lagrangian, the Minkowski metric could be used as a special case. According to Hoffmann this was no longer possible for his Lagrangian because “the electromagnetic field exerts a gravitational influence” ([275 ], p. 364). As he could not find a solution to his complicated field equations, the “degree of modification of the electrostatic potential by its own gravitational field” could not be determined.

In connection with the work of Euler and Kockel on the scattering of light by light under his guidance, W. Heisenberg wrote Pauli on 4 November 1934: “The terms to be added to the Lagrangian look like in the theory of Born and Infeld, but they are twenty times larger than those of Born and Infeld” ([488 *], p. 358).⁷¹ But Pauli had not changed his opinion; in connection with the scattering of light by light, he answered Heisenberg curtly: “I do not care about Born’s theory” ([488 ], p. 372). Ten years later, in his letter to Einstein of 10 October 1944 Born assessed his theory with some reservation ([168 *], p. 212):

“[…] I always had a lot of understanding for your good Jewish physics, and much amusement with it; however, I myself have produced it only once: the non-linear electrodynamics, and this is no particular success […].”⁷²

Nevertheless, it had some influence on UFT; cf. Sections 6.1.3, 9.7, and 10.3.4.

Born and Infeld unsuccessfully tried to quantize their non-linear theory of the electromagnetic field by using the commutation rules of Heisenberg and Pauli for the field strenghts [43 , 44 ]. They noticed that the theory could be presented differently according to whether the pairs $→ → E, B$ , or $→ → D, H$ ; $→ → D, B$ ; $→ → E, H$ were chosen as independent variables. The authors took $→ → D, B$ in order to avoid “formal difficulties”. However, a perturbative approach by canonical quantization of either the field or the vector potential could not succeed because the interaction term in the Hamiltonian included higher powers of derivative terms.⁷³

One who became attracted by the Born–Infeld theory was E. Schrödinger. He had come “across a further representation, which is so entirely different from all the aforementioned, and presents such curious analytical aspects, that I desired to have it communicated” ([542 ], p. 465). He used a pair $ℱ ,𝒢$ of complex combinations of the 3-vector fields $→ → → → B, E, H, D$ such that $→ → → → ℱ = B − iD, 𝒢 = E + iH$ . The Lagrangian $ℒ$ was to be determined such that its partial derivatives with respect to $ℱ$ and $𝒢$ coincided with the complex conjugates: $ℱ¯ = ∂ℒ- ∂𝒢$ and $𝒢¯= ∂ℒ- ∂ℱ$ . The result is

Born’s constant $b$ was set equal to one. Schrödinger showed that his formulation was “entirely equivalent to Born’s theory” and did not provide any further physical insight. Thus, Schrödinger’s paper gave a witty formal comment on the Born–Infeld theory. Ironically, it had been financed by Imperial Chemical Industries, Limited.

S. Kichenassamy⁷⁴ studied the subcase of an electromagnetic null field with matter tensor: $2 i Tij = A kikj, kik = 0$ and showed that in this case the Born–Infeld theory leads to the same results as Maxwell’s electrodynamics [328 , 340 ].

	Abstract
1	Introduction
2	Mathematical Preliminaries
	2.1	Metrical structure
	2.2	Symmetries
	2.3	Affine geometry
	2.4	Differential forms
	2.5	Classification of geometries
	2.6	Number fields
3	Interlude: Meanderings – UFT in the late 1930s and the 1940s
	3.1	Projective and conformal relativity theory
	3.2	Continued studies of Kaluza–Klein theory in Princeton, and elsewhere
	3.3	Non-local fields
4	Unified Field Theory and Quantum Mechanics
	4.1	The impact of Schrödinger’s and Dirac’s equations
	4.2	Other approaches
	4.3	Wave geometry
5	Born–Infeld Theory
6	Affine Geometry: Schrödinger as an Ardent Player
	6.1	A unitary theory of physical fields
	6.2	Semi-symmetric connection
7	Mixed Geometry: Einstein’s New Attempt
	7.1	Formal and physical motivation
	7.2	Einstein 1945
	7.3	Einstein–Straus 1946 and the weak field equations
8	Schrödinger II: Arbitrary Affine Connection
	8.1	Schrödinger’s debacle
	8.2	Recovery
	8.3	First exact solutions
9	Einstein II: From 1948 on
	9.1	A period of undecidedness (1949/50)
	9.2	Einstein 1950
	9.3	Einstein 1953
	9.4	Einstein 1954/55
	9.5	Reactions to Einstein–Kaufman
	9.6	More exact solutions
	9.7	Interpretative problems
	9.8	The role of additional symmetries
10	Einstein–Schrödinger Theory in Paris
	10.1	Marie-Antoinette Tonnelat and Einstein’s Unified Field Theory
	10.2	Tonnelat’s research on UFT in 1946 – 1952
	10.3	Some further developments
	10.4	Further work on unified field theory around M.-A. Tonnelat
	10.5	Research by and around André Lichnerowicz
11	Higher-Dimensional Theories Generalizing Kaluza’s
	11.1	5-dimensional theories: Jordan–Thiry theory
	11.2	6- and 8-dimensional theories
12	Further Contributions from the United States
	12.1	Eisenhart in Princeton
	12.2	Hlavatý at Indiana University
	12.3	Other contributions
13	Research in other English Speaking Countries
	13.1	England and elsewhere
	13.2	Australia
	13.3	India
14	Additional Contributions from Japan
15	Research in Italy
	15.1	Introduction
	15.2	Approximative study of field equations
	15.3	Equations of motion for point particles
16	The Move Away from Einstein–Schrödinger Theory and UFT
	16.1	Theories of gravitation and electricity in Minkowski space
	16.2	Linear theory and quantization
	16.3	Linear theory and spin-1/2-particles
	16.4	Quantization of Einstein–Schrödinger theory?
17	Alternative Geometries
	17.1	Lyra geometry
	17.2	Finsler geometry and unified field theory
18	Mutual Influence and Interaction of Research Groups
	18.1	Sociology of science
	18.2	After 1945: an international research effort
19	On the Conceptual and Methodic Structure of Unified Field Theory
	19.1	General issues
	19.2	Observations on psychological and philosophical positions
20	Concluding Comment
	Acknowledgements
	References
	Footnotes
	Biographies