Einstein and the Quantum (24 page)

Throughout all this change, the constant in Einstein's life was his scientific work, and in particular his focus on what he considered the most important problem of the day, quantum theory. Nernst's visit to see Einstein in Zurich, and Nernst's high-profile focus on quantum problems, had finally begun to make the physics community aware of what Einstein had known since 1907: there were really two puzzles of quantum theory. The radiation formula, photoelectric effect, and other similar phenomena suggested that light, conventionally conceived as a wave, came in quantized energy units,
hυ
, and had particulate properties. At the same time the specific heat theory, now corroborated in its essentials by Nernst's experiments, pointed strongly to the proposition that molecular mechanics violated Newton's laws and also involved quantization of energy in units of
hυ
, at least when the molecular motion was periodic. Since these vibrations could be electrically neutral and thus not interact with light directly, this behavior appeared to be
essentially independent of the quantum properties of radiation. Einstein had a major decision to make: which quantum problem would he focus on? He chose the quantum theory of radiation.

Already in early 1909 Einstein had confided to his idol, Lorentz, that his program was to tinker with Maxwell's equations in order to generate a theory of light quanta. Lorentz, who undoubtedly had the deepest understanding of electromagnetic theory of his generation, had warned Einstein, “
as soon as one makes even the slightest
change in Maxwell's equations, one is faced … with the greatest difficulties.” As he toiled onward through 1909 and 1910, Einstein began to realize that Lorentz was prescient; changing Maxwell's equations was like touching up the
Mona Lisa
. Not frivolously had Boltzmann proclaimed of these equations, “was it God that wrote those lines?” The mathematical structure was perfect, and every modification implied a contradiction with the multitude of known electromagnetic phenomena. Einstein was in the midst of these difficult explorations in September of 1909 when he admitted in the conclusion of his Salzburg lecture, “
it has not been possible to formulate
a mathematical theory of radiation which will do justice both to the undulatory structure and the … quantum structure…. [The] fluctuation properties of radiation [that he had demonstrated in his lecture] … offer few formal clues on which to build a theory.” He boldly prophesied the emergence of a fusion theory of radiation, yet had nothing to show in support but his vague notion of “singularities” attached to an extended field, which he quickly qualified with the statement, “no importance should be attached to such a picture as long as it has not led to an exact theory.”

In his correspondence throughout the year after Salzburg lecture, Einstein repeatedly alludes to his struggles with radiation theory. On New Year's Eve of 1909 he writes again to Laub, who was his constant sounding board during this period: “
I have not yet arrived at solution
to the light quanta question, but I have found quite a few significant things while working at it. I'll see whether I might not yet succeed in hatching this favorite egg of mine.” In January of 1910 in a letter to Sommerfeld he describes his increasingly radical hypotheses to explain the dual properties of radiation: “
maybe the electron is not
to be
conceived as such a simple structure as we think? There is nothing one would not consider when one is in a predicament.” By July of that year, buoyed by Nernst's visit, he writes again to Sommerfeld, confident that the two quantum hypotheses (for light and for matter) are correct but allowing that no progress has been made reconciling quanta and waves. He has decided that “
[a] crudely materialistic conception
of the point structure of radiation … cannot be worked out.” On August 2, to Laub, shortly after being visited by Sommerfeld, he writes: “
I have not made any progress
regarding the question of the constitution of light. There is something very fundamental at the bottom of it.” Then, again to Laub, in November of 1910, a ray (or wave?) of light: “
At the moment I am very hopeful
that I will solve the radiation problem, and that I will do so without light quanta. I am awfully curious how the thing will turn out. One would have to give up the energy principle [conservation of energy] in its current form.” Alas, one week later: “
The solution of the radiation problem
has again come to naught. The devil played a dirty trick on me.”

Unfortunately, none of the notes containing Einstein's failed attempts to quantize radiation have survived, and we know very little about the mathematical structures he played with and discarded during this period. This letter is, however, the last time that Einstein suggests he knows how or is close to solving the quantum puzzle for radiation. In December of 1910, again he expresses frustration: “
the riddle of radiation
will not yield … the secret remains unsolved.”

In hindsight we know that Einstein was on the wrong track in trying to change Maxwell's equation to encompass quanta. Einstein was trying to do the most natural thing: find a new set of electromagnetic equations that contained the fundamental quantum constant,
h
, either explicitly or perhaps, he thought, implicitly, through the ratio
e
²
/
c
(the electron charge squared divided by the speed of light), which has the same units as Planck's constant. All the other known constants of nature at the time (except
h
and
e
) appeared in the known laws of nature. The speed of light appears directly in Maxwell's equations; the gravitational constant,
G
, in the Newtonian law of gravity; Boltzmann's constant,
k
, in the entropy principle,
S
=
k
log
W
. As these constants
appear in the fundamental laws, one can calculate from these laws consequences that depend on the values of
c
or
G
or
k
.

Planck's constant was not introduced in connection with a new physical law; its introduction was an ad hoc insertion in the midst of evaluating the blackbody entropy from Boltzmann's principle. Einstein's hypothesis of light quanta had the same ad hoc character, essentially inherited from Planck. That is why Einstein repeatedly emphasized that “
the so-called quantum theory of today
is, indeed, a helpful tool but … it is not a theory in the usual sense of the word.” He naturally assumed that a quantum theory of radiation would require a generalization of Maxwell's equations, which would introduce Planck's constant but would reduce to the original equations in contexts where no hint of quantum behavior was observed.

But the deck was stacked against him, and his assumption was wrong for a most subtle reason. Maxwell's equations remain true and unchanged in quantum theory; it is only their
interpretation
that changes: they are the
wave
equation governing the dynamics of the photon, in the same sense that eventually the Austrian physicist Erwin Schrödinger would discover the quantum wave equation describing the dynamics of the electron. But there is a decisive difference between the two equations: right in the middle of Schrödinger's equation for the electron, planted like a flag, is Planck's constant, just as Einstein had expected in an analogous equation for photons. Einstein had picked the short straw. In the radiation problem the “quantum of action,”
h
, was hidden in plain sight, invisible because of the principle of relativity, Einstein's own creation!

If there were indeed particles of light, they could have no inertial mass, as no massive object can reach the full speed of light. Nonetheless, as we have seen, radiation can exert pressure (i.e., it can transfer momentum) even though it is massless.
³
The relativistic relation between the energy,
E
, of a light wave and its momentum,
p
, is
E
=
pc
; this relationship is embedded in Maxwell's differential equations. In
quantum theory, as we now understand it, photon energy is quantized and proportional to
h
,
but so is momentum
; so the factor
h
cancels in the equation relating the two. This is the essential reason that
h
does not appear in Maxwell's equations. It “should” be there, but it cancels because photons are massless. This does not happen for massive particles, which thus are governed by quantum equations in which
h
sticks out like a sore thumb. We now believe that photons are the
only
freely propagating massless particles in our universe, so Maxwell's equations are the only quantum equations where
h
does not appear explicitly. How's that for bad luck? If Einstein had instead decided to focus on the mechanics of electrons in atoms, as would Niels Bohr in a just a couple of years, perhaps the history of physics would have been different. But his quixotic search for the quantum version of Maxwell's equations defeated him, and he soon would lay down his lance.

By May of 1911, writing from Prague, a new note of resignation appears in a letter to Besso. “
I no longer ask whether these quanta
really exist. Nor do I try to construct them any longer, for I now know that my brain cannot get through in this way. But I rummage through the consequences as carefully as possible so as to learn the range of applicability of this conception.” And then, just as Einstein is abandoning his four-year struggle, comes the invitation:

Dear Sir,

To all appearances
, we are at the moment in the midst of new developments regarding the principles on which the classical molecular and kinetic theory of matter has been based…. Messrs. Planck and Einstein have demonstrated that … contradictions disappear if one imposes certain limitations on the movement of electrons and atoms, … but this interpretation in turn … would necessarily and indisputably entail a vast reform of our current fundamental theories.

To that end, the undersigned proposes to you to participate in a “Scientific Congress” which will … bring together in a small meeting, several eminent scientists…. I hope that I can count on your collaboration, and I beg to assure you, dear, Sir, of my highest esteem.

Signed: Ernest Solvay, June 9, 1911

One can just imagine Einstein's reaction as he read the letter: Tell me about it, buddy. The invitation was of course too prestigious to turn down, and he feigns great interest in his reply to Nernst a few days later. He has been asked to give the report on the current status of the problem of specific heat, and he accepts. But his heart, and his fertile scientific imagination, are elsewhere. By August 1911 his correspondence with Laub contains the first hints of a new passion: “
The relativistic treatment of gravitation
is causing serious difficulties. I consider it probable that … the principle of the constancy of the velocity of light holds only for spaces of constant gravitational potential.” Within two weeks of that letter he is corresponding with the astronomer Willem Julius about the redshift of the wavelength of light rays from the sun due to its gravitational
⁴
field. In September, with the Solvay Congress looming in less than two months, he replies irritably to a letter from Besso: “
If my answer is not … thorough
, it's because my drivel for the Brussels congress weighs down on me.”

Thus, while the First Solvay Congress was a point of departure for the field, the quantum problems it addressed would henceforth be the central problems of physics, it marked a temporary surrender for its youngest participant. Einstein's report was thorough and scholarly and explained how sharply contradictory the various modes of reasoning were, as revealed, for example, by the fluctuations in the energy of radiation: “
We stand here before an unsolved puzzle
, just as in the study of thermal motion in a solid…. Who would have the audacity to give a categorical answer to these questions? I only intended to show here how fundamental and deep-rooted the difficulties are in which the radiation formula enmeshes us.” He presented no new ideas for how to get out of these difficulties; the optimism of Salzburg had dissipated.

On a personal level Einstein enjoyed the conference greatly; he described how he was “enchanted” with the French trio of Jean Perrin, Paul Langevin, and Madame Curie. Lorentz, whom he had already
met for the first time earlier that year, again awed him: “
H. A. Lorentz chaired the conference
with incomparable tact and unbelievable virtuosity.” Planck's integrity won him over: “
he is a completely honest man
who shows no consideration for himself.” But as for science, everyone there was just discovering the forbidding territory he had been surveying for years. His final verdict was delivered to his old friend Besso, for whom diplomacy was not required: “
In Brussels, too
, they acknowledged the failure of the theory … but without finding a remedy. In general the Congress … resembled the lamentations on the ruins of Jerusalem. Nothing positive has come out it…. I did not find it very stimulating because I heard nothing that I had not known before.” In the aftermath he seems to have suspected that his focus on radiation was the wrong path, writing to Lorentz a month later, “
the
h
-disease
⁵
looks ever more hopeless.
Still I believe that the purely mechanical side will be the first to be cleared up
.” In this he would be proved right.

There was one claim enunciated at Brussels that Einstein alone of all present might have found particularly interesting and worthy of dispute, although he would never have done so in public, because that claim was made by Solvay himself. In the midst of his incomprehensible lecture on “positive and negative ether” Solvay had included the statement, “
I took as my starting point
Newton's wonderful law [of gravity], which is uncontested and therefore able to satisfy the most rigorous scientific mind.” In fact it was just this law that Einstein had now begun to question seriously. By March of 1912 he wrote to a friend, “
I'm working at full speed
on a problem (gravitation). You should forgive me my long silence.” In October of 1912, having returned to Zurich, he declined an invitation from Sommerfeld to speak on quantum theory with the words, “
I assure you that I have nothing
new to add to the question of quanta that might be of any interest…. I am now working exclusively on the gravitation problem.” Sommerfeld, who had contacted Einstein with the speaking invitation on behalf of the famous mathematician David Hilbert, wrote in despair to
Hilbert, “
My letter to Einstein proved useless
…. Apparently he is so deeply involved in the problem of gravitation that he turns a deaf ear to all else.”