Vector Potential, Gauge Field and Connection on a Fiber Bundle
Chen Ning Yang
Institute for Theoretical Physics, State University of New York
Stony Brook, New York 11794-3840, USA
A number of basic concepts in contemporary physics have deep relations with each other. In this talk I shall trace the early origins of some of these concepts.
1 Vector Potential
We shall start with Maxwell (1831-1879). Maxwell wrote three papers^{[1-3]} in 1855-1865 which founded the theory of electromagnetism. When he started on these works soon after his degree [see letter from Maxwell to W. Thomson dated February 20, 1854, reprinted in Proc. Camb. Phil. Soc. 32(1936) 695], the first three of the four EM laws: Coulomb’s law, Gauss’ law and Ampere’s law were already known to be expressible in differential form. But Faraday’s law had not yet been so expressed. Borrowing from W. Thomson’s (1824-1907) earlier papers on magnetism where the vector potential was introduced (called three functions F, G and H), Maxwell in paper 1 identified this vector potential with Faraday’s intuitive idea of an “electro-tonic state”. Near the end (but before the “examples”) of this paper he summarized “the Theory of the Electro-tonic State” in six laws. The last of these laws was, in today’s notation,
,
which is Faraday’s law.
That Maxwell regarded his interpretation of Faraday’s intuitive idea of an electro-tonic state (which he sometimes called electro-tonic intensity) as an important discovery is evident from the following passage at the end of paper 1, the first of his three papers on electromagnetism.
With respect to the history of the present theory, I may state that the recognition of certain mathematical functions as expressing the “electro-tonic state” of Faraday, and the use of them in determining electro-dynamic potentials and electro-motive forces is, as far as I am aware, original; but the distinct conception of the possibility of the mathematical expressions arose in my mind from the perusal of Prof. W Thomson’s papers...
The main contribution of Maxwell’s second paper^{[2]} was the introduction of the displacement current. This introduction curiously occurred in Part 3 of this paper where Maxwell wrote:
Prop XIV ¾ To correct the equation (9) (of Part 1) of electric currents for the effect due to the elasticity of the medium.
Precisely how Maxwell arrived at the correction term is unclear to me. The paper had begun with the following declaration:
My object in this paper is to clear the way for speculation in this direction, by investigating the mechanical results of certain states of tension and motion in a medium, and comparing these with the observed phenomena of magnetism and electricity.
There followed a “Theory of Molecular Vortices applied to Electric Currents” with beautifully intricate lattices of vortex motion. The rambling style of discourse is very difficult to follow and I failed^{[4]} to penetrate the detailed steps that had led Maxwell to the “correction”. What was clear was that it was not a result of his discovering that electric charge would not be conserved without the correction ¾ a myth that is continually perpetuated. (Of course, after making the correction, Maxwell immediately did derive what he called the equation of continuity [of electricity].)
With the introduction of the displacement current, Maxwell’s equations are essentially complete. In the next paper, paper 3, he returned to these equations, collecting and tabulating them. It is especially noteworthy that he emphasized in Section (73):
I have on a former occasion* attempted to describe a particular kind of motion and a particular kind of strain, so arranged as to account for the phenomena. In the present paper I avoid any hypothesis of this kind; and in using such words as electric momentum and electric elasticity in reference to the known phenomena, of the induction of currents and the polarization of dielectrics, I wish merely to direct the mind of the reader to mechanical phenomena which will assist him in understanding the electrical ones. All such phrases in the present paper are to be considered as illustrative, not as explanatory.
*Here Maxwell referred to his paper 2 (note by C. N. Y.).
In other words, he now disavowed the relevance of the mechanical strain (tension) that was the main topic of investigation in paper 2. Also noteworthy was the following words in Section (74):
In speaking of the Energy of the field, however, I wish to be understood literally. All energy is the same as mechanical energy, whether it exists in the form of motion or in that of elasticity, or in any other form. The energy in electromagnetic phenomena is mechanical energy. The only question is, where does it reside? On the old theories it resides in the electrified bodies, conducting circuits, and magnets, in the form of an unknown quality called potential energy, or the power of producing certain effects at a distance. On our theory it resides in the electromagnetic field, in the space surrounding the electrified and magnetic bodies, as well as in those bodies themselves, and is in two different forms, which may be described without hypothesis as magnetic polarization and electric polarization, or, according to a very probable hypothesis as the motion and the strain of one and the same medium.
Here the fundamental spirit of the concept of the field, so dominant in twentieth century physics, was for the first time clearly and emphatically enunciated.
Throughout the last few decades of the nineteenth century, physicists considered Maxwell’s papers confusing and difficult. This is hardly surprising since Maxwell’s papers and his book^{[5]} were not well written. A simplification of the equations was discovered by Hertz (1857-1894) and Heaviside (1850-1925) independently: From Maxwell’s equations the vector and scalar poten-tials and Ψ can be altogether eliminated, resulting in the four compact vector equations for the four EM laws that are still found in all textbooks on EM today. Heaviside, a brilliant engineer, was very happy with this simpli-fication of Maxwell’s equations. He wrote^{[6]} that it brought
to light interesting relations which were formerly hidden from view by the intervention of the vector potential A, and its parasites J and Ψ.
Whittaker agreed^{[7]} enthusiastically with Heaviside:
The great service which Heaviside now rendered to science was to clear away this accumulation of rubbish...
However, the vector and scalar potentials do have measurable meaning in quantum mechanics, and should not be completely eliminated, as we shall discuss later.
2 Gauge Symmetry
It is of course transparent that Maxwell’s equations have profoundly influenced most aspects of twentieth century technology. What may not have been sufficiently emphasized is that Maxwell’s equations, through its two symmetries, have also profoundly shaped basic theoretical physics in this century. These symmetries are Lorentz symmetry and gauge symmetry.
Lorentz symmetry is not a subject matter of my talk today. Gauge symmetry was first proposed in 1918 by Weyl (1885-1955) in response to Einstein’s (1879-1955) call for a geometrical theory which would embrace both general relativity and EM. I had elsewhere^{[8]} discussed in some detail the origin of Wely’s ideas and shall not repeat it here. Suffice it to mention that Weyl had sought to generalize the idea of parallel displacement in general relativity. If a vector transported around a closed loop back to its original position could change its direction, he asked, “Why not also its length?”^{[9]} He thus proposed a length changing (i.e., scale changing) factor along a path:
. (1)
Weyl’s length changing idea was rejected by Einstein who pointed out^{[8]} that it implied the impossibility of establishing a standard for meter sticks. The idea would certainly have died out were it not for the fact that when quantum mechanics was developed in 1925-1927, it was found by Fock (1898-1974) and by F. London (1900-1954) that quantum mechanics required the insertion of an in Weyl’s theory, so that length change became a phase change:
. (2)
Weyl then came back in 1929 and rewrote his idea, embracing the phase changing factor (2), thereby side-stepping Einstein’s objection.
3 Converting Requirements of Gauge Symmetry
into an Active Principle for Writing Down
Interactions
Weyl’s gauge symmetry was well known to physicists in the 1940’s, mostly from Pauli’s review article in the Handbuch der Physik, volume 24. The symmetry was used to check results of calculations. In fact, when I was a graduate student in 1946-1948, and a postdoc in 1948-1950, young theorists all knew that it was “smart” to ask at the end of a theoretical seminar whether the result was gauge invariant. One may call that a passive use of gauge symmetry.
In 1954 Mills and I converted this passive use into an active use. In an abstract^{[10]} we wrote
The conservation of isotopic spin points to the existence of a fundamental invariance law similar to the conservation of electric charge. In the latter case, the electric charge serves as a source of electromagnetic field; an important concept in this case is gauge invariance which is closely connected with (1) the equation of motion of the electromagnetic field, (2) the existence of a current density, and (3) the possible interactions between a charged field and the electromagnetic field. We have tried to generalize this concept of gauge invariance to apply to isotopic spin conservation. It turns out that a very natural generalization is possible.
A different motivation for the generalization was given in another paper:^{[11,12]}
The differentiation between a neutron and a proton is then a purely arbitrary process. As usually conceived, however, this arbitrariness is subject to the following limitation: once one chooses what to call a proton, what a neutron, at one space-time point, one is then not free to make any choices at other space-time points.
It seems that this is not consistent with the localized field concept that underlies the usual physical theories. In the present paper we wish to explore the possibility of requiring all interactions to be invariant under independent rotations of the isotopic spin at all space-time points, so that the relative orientation of the isotopic spin at two space-time points becomes a physically meaningless quantity...
The result of these considerations can be summarized^{[13]} by saying that the phase factor (2) is generalized to become an element of a Lie group:
(3)
where the product is an ordered product, an element of a Lie algebra, and is a generalization of the vector potential .
4 Aharonov-Bohm Experiment
After the 1929 paper of Weyl about gauge fields (phase fields actually), no one seemed to have returned to Einstein’s objection to the original idea of Weyl’s about length changing. Neither Einstein nor Weyl did. Also not London or Fock or Pauli. Had one of them done that, he would certainly have concluded that the phase difference between two meter sticks do not affect their lengths, thus rendering Einstein’s original objection inoperative. But he might further ask whether the phase difference is measurable. Had that question been raised, it is likely that the now famous Aharonov-Bohm experiments would be proposed thirty years earlier.
As it was, when Aharonov and Bohm proposed^{[14,15]} their experiments in 1959, they were apparently unaware of Einstein’s objection in 1918 to Weyl’s original idea about length changing. What they pointed out was an intrinsic property of EM, an Abelian gauge theory, showing that the vector and scalar potentials do have measurable consequences in quantum mechanics. The experiments were difficult to do. Furthermore the generations-old dogma, which had started with the work of Hertz and of Heaviside mentioned above, that the vector and scalar potentials were not physically meaningful (rubbish, in Whittaker’s words), but were only auxiliary concepts useful for calculation, had created great resistance toward assigning any physically meaningful role to these potentials. The final definitive magnetic Aharonov-Bohm experiment, utilizing the new technology of electron holography, was done by Tonomura and collaborators^{[16]} in their beautiful experiments of 1982 and 1986.
5 Vector Potential as Connection on a Fiber
Bundle
Mathematicians have studied the abstract concept of connections on fiber bundles since the 1940’s. A number of scattered papers in the 1960’s and 1970’s have pointed out that gauge theories are related to this concept. However, these papers did not create much impact either in mathematics or in physics. Then in 1975, T. T. Wu and I examined,^{[17]} through the Aharonov-Bohm experiment and Dirac’s monopole theory, the intrinsic meaning of EM in quantum mechanics. The topological implications of EM^{[18]} became apparent and the description of gauge fields in terms of connections on fiber bundles, with all its topological connotations, became generally accepted.
6 Origins of Wave Mechanics
In 1926 in the epoch-making papers that created wave mechanics, Schrodinger (1887-1961) did not refer to a paper^{[19]} that he had published in 1922. This earlier paper was largely forgotten until it was discovered by two historians of science, Raman and Forman,^{[20]} in a paper with the interesting title “Why was it Schrodinger who developed de Broglie’s idea?”. These authors pointed out that Schrodinger had come very close in 1922 to the core concepts of wave mechanics, and needed only a nudge, from de Broglie’s proposal, to create in 1926 his monumental work of 1926. Raman and Forman’s thesis was later proven by Hanle^{[21]} who discovered a letter dated November 3, 1925 from Schrodinger to Einstein:
The de Broglie interpretation of the quantum rules seems to me to be related in some ways to my note in the Zs. f. Phys. 12, 13, 1922, where a remarkable property of the Weyl “gauge factor” exp [- ò f dx] along each quasi-period is shown. The mathematical situation is, as far as I can see, the same, only from me much more formal, less elegant and not really shown generally. Naturally de Broglie’s consideration in the framework of his large theory is altogether of far greater value than my single statement, which I did not know what to make of at first.
Schrodinger’s 1922 paper was entitled “On a Rmarkable Property of the Quantum Orbits of a Single Electron”. It started with Weyl’s length changing factor (1) which he called Streckenfaktor. He wrote^{[19]} it in the following form:
. (4)
He then calculated the exponent for a Bohr orbit, (remember this was before 1925!) and found it “remarkable” that the exponent was an integral multiple of .
In retrospect what is even more remarkable is the few paragraphs near the end of this 1922 article, where Schrodinger toyed with the idea of writing
.
Had he pursued this idea in depth, quantum mechanics might have been born three years before 1925! [cf. further discussions of Schrodinger’s 1922 paper in my article in Schrodinger, Centenary Celebration of a Polymath, Ed. C. W. Kilmister (Cambridge University Press, 1987)].
7 Further Subtleties in Space-Time Structure?
It is truly astonishing (cf. Table 1) that Faraday’s vague intuitive non-mathematical idea of the electro-tonic state has metamorphosed into the sophisticated and precise mathematical concept of the connection on a fiber bundle. This remarkable development provides a good example of how experimental physics, theoretical physics and mathematics can come together to reveal a fundamental structure of the physical universe.
A crucial step in this development was the expression
(classical). (5)
Maxwell did recognize explicitly in 1865 that A is a momentum (see Table 1). He also recognized explicitly that A is not uniquely defined in . How then did he reconcile this flexibility in A with the non-flexible concept of momentum? I looked into his papers and his book but did not find the answer.
We now know that in quantum mechanics expression (5) is replaced by
(quantum mechanics) (6)
which has the property that the flexibility in A is compensated for by the corresponding change in when a phase factor is introduced in. That is in fact the gauge principle.
That this compensation is related to the differential structure of space-time is shown by the fact that (6), or
(7)
is the covariant derivative of .
We thus come to the partial realization of an idea first stated by Einstein in 1934 in an article^{[22]} with the title The Problem of Space, Ether, and the Field in Physics:
But the idea that there exist two structures of space independent of each other, the metric-gravitational and the electromagnetic, was intolerable to the theoretical spirit. We are prompted to the belief that both sorts of field must correspond to a unified structure of space.
Table 1 Name and Notation for
Name |
Notation |
Source |
3 functions Electro-tonic state |
F, G, H a _{0, b 0, g 0} |
Kelvin 1851 Maxwell 1855-1856 (p.202, SP vol.) |
Electro-tonic state |
F, G, H |
Maxwell 1861-1862 (p.476, SP vol. ) |
Electromagnetic momentum |
F, G, H |
Maxwell 1865 (p.561, SP vol. ) |
Electromagnetic momentum (§618) |
F, G, H |
Maxwell 1873 (Treatise) |
Rubbish |
Heaviside 1893 Whittaker 1928 | |
Connection on Fiber Bundle |
～1975 |
SP=Scientific Papers of J. C. Maxwell, ed. W. D. Niven(1927)
References