© 2014 by Fernando Caracena

**Initial Conditions**—**A bit of ancient history**

**Initial Conditions**

**A bit of ancient history**

"There is nothing new to be discovered in physics now. All that remains is more and more precise measurement."

—William Thomson a.k.a. Lord Kelvin (1824-1907)

The future work in physics at the end of the 19^{th} Century was portrayed by a leader in the field as boring! Only two dark clouds hung over physics

"The beauty and clearness of the dynamical theory, which asserts heat and light to be modes of motion, is at present obscured by two clouds. I. The first came into existence with the undulatory theory of light, and was dealt with by Fresnel and Dr Thomas Young; it involved the question, How could the earth move through an elastic solid, such as essentially is the luminiferous ether? II. The second is the Maxwell-Boltzmann doctrine regarding the partition of energy."

--

Lord Kelvin, Nineteenth Century Clouds over the Dynamical Theory of Heat and Light, Philosophical Magazine, Sixth Series, 2, 1–40 (1901).

Clarifying the two dark areas of physics became the main task of physics most of the 20^{th} Century. The first dark cloud concerned the paradox the issue of the motion of the Earth through a luminiferous aether. Waves require a medium to propagate through, we should be able to detect our motion through that medium by some clever experiment. However, Maxwell's theory of electromagnetic waves indicates that the speed of light in a vacuum is a fundamental constant. But wait, we know that everything in the Universe is in motion! How can the speed of light be the same in all those frames of reference? In orbiting the Sun, the Earth moves along a big elliptical path. On it, if light moves through some kind of medium at a fundamental speed, then sometimes the Earth moves through that medium in one direction, then other times, in the opposite direction. For example, this happens over twenty four hours everyday, because each point on the Earth goes around in a circle each day because of the Earth's spin. However, the Michaelson Morley experiment (published in 1887) performed by professors of the graduate school I attended (Case Western Reserve University) failed to detect any motion of the Earth through space. The inescapable conclusion was that there was no *luminiferous* ether. Still, the speed of light was a universal constant according to Maxwell's well-confirmed theory of electro-magnetism. But how could that be? This presented a major conundrum. An unconventionally thinking Jewish genius solved this problem. Albert Einstein, who had been considered a slow learner by his grammar school teachers (see "Prelude to Special Relativity"), solved a major problem that had stumped the greatest minds in the world. Eventually, this man would later receive a Nobel prize, but strangely, not by performing this outstanding feat.

Albert Einstein was the very deep and imaginative thinker that was able to solve many of the important problems in the physics of his time. He helped to clarify both the cloudy areas identified by Lord Kelvin, but his biggest contribution was in the area of the propagation of light, in the process of which he come up with a new theory, the Theory of Relativity. Additionally, his work led him to generalize the work of Newtonian gravity into a theory of warped space-time in the General Theory of Relativity. See "Special Relativity II—standard notation".

*Max Planck *uncovered the tools for grappling with one of the dark clouds over physics. Using an elaborate bookkeeping scheme, he stumbled upon the solution to the puzzle of Black-body radiation, which resulted in the quantum theory. (See "Prelude to Quantum Physics"). He solved the problem by letting each normal mode of vibration of light in a cavity have an energy proportional to its frequency of oscillation, which is known by the famous formula

E = h f.

Through this method he was able to treat the energy of an extended wave structure as if it were a particle.

At the end of the calculation, Planck had planned to find the limit of the energies involved as the constant of proportionality, h, approached zero (h —>0). But he found that this was not possible to do. The only way to make his elaborate calculations work was to solve for h as a constant that was consistent with measurements of black-body spectra. This solution stuck; it was consistent with all experimental data; and so he published the result. The conclusion was that h is a universal and fundamental constant. Continuing to ponder in bewilderment over the crazy results, he did not contribute much more to the Quantum Theory in his later years.

Einstein again contributed to pushing forward the frontier of physics by clarifying the nature of the Planck hypothesis.In analysing the photoelectric effect using the Quantum hypothesis, he treated light as particles, discrete bundles of energy, having both momentum and spin. There was no need for a luminiferous medium, through which light travels. See "Photons and the Electromagnetic field".

Niels Bohr, a Danish physicist, was perhaps the main driving spirit behind the development of the Quantum Theory. He was not so much a mathematical theorist, but more of a philosopher who influenced other premier theorists, Like Werner Heisenberg, the author of the matrix theory of quantum mechanics. Bohr's own efforts resulted only in the Old Quantum theory; however, philosophically he continued to interpret the quantum theory in a way that guided its development.

Perhaps the strongest contributer to the theoretical development of the Quantum Theory was Erwin Schrödinger who proposed the famous wave equation that described the motion of quanta in terms of probability amplitudes. (See the blog, "The Schrödinger equation"). Bohr and company had tasked themselves with finding an interpretation of the Schrödinger equation, and the meaning of its wave function. After some work, quantum theorists found out that this equation was equivalent to Heisenberg 's matrix mechanics. Bohr subsequently interpreted the wave function was as a complex probability amplitude for detecting a single particle at any point in its trajectory. Bohr's and his associates' ideas about quantum theory were called the Copenhagen interpretation.

Einstein clashed many times with Niels Bohr at major scientific meetings, in a series of debates over the meaning of Quantum Theory.

"Einstein was displeased with quantum theory and mechanics (the very theory he helped create), despite its acceptance by other physicists, stating that God "is not playing at dice."

^{[129]}Einstein continued to maintain his disbelief in the theory, and attempted unsuccessfully to disprove it until he died at the age of 76.^{[130]}In 1917, at the height of his work on relativity, Einstein published an article inPhysikalische Zeitschriftthat proposed the possibility of stimulated emission, the physical process that makes possible the maser and the laser.^{[131]}This article showed that the statistics of absorption and emission of light would only be consistent with Planck's distribution law if the emission of light into a mode with n photons would be enhanced statistically compared to the emission of light into an empty mode."

Einstein felt that Quantum Theory was yet incomplete, therefore needing further development. Although some physicist feel that Niels Bohr won the debates, Einstein put a serious doubt in the minds of physicists over what was really happening behind the scenes in quantum phenomena. Eventually, physicists dealt with the quantum quandaries by just ignoring them. "Shut up and calculate!" became the new slogan in physics.

""If I were forced to sum up in one sentence what the Copenhagen interpretation says to me, it would be '

Shut up and calculate!'

Some physicists have modified the "Shut-up-and-calculate" philosophy to a more softened one, "Calculate first and discuss the interpretation afterwards".

*Contributions of the 20*^{th} Century

*Contributions of the 20*

^{th}CenturyUnfortunately, even if they did "Shut up and calculate!", physicists hit some trouble spots in trying to avoid theoretical instabilities and infinities.

Paul Dirac, a very private and colorful person who is sometimes described by biographers as borderline autistic ("The Strangest Man: The Hidden Life of Paul Dirac, Mystic of the Atom" by Graham Farmelo, 2011, Basic Books, NY ), came up with a relativistic generalization of the Schrödinger equation.

Although it gave better agreement with spectral results than the Schrödinger Equation, Dirac 's Equation of the Electron had a huge instability that one had to ignore to "Shut up and calculate!" The occurrence of negative energy states in the solution of Dirac's Equation allowed electrons to cascade into an infinitely deep energy well, which made them basically unstable particles, each one capable of releasing an infinity of energy. To this, Dirac came up with a clever solution: just consider that the physical vacuum has all the negative electron states filled with electrons. Since electrons are fermions, this means that a free electron cannot fall into already occupied negative energy states, so it remains stable.

Dirac's facile solution to the vacuum instability of an electron fixed the problem with his equation, but physicists began to realize what it implied: there cannot be a theory of a single electron. Dirac 's theory of the electron became a theory that involved an infinite number of electrons, all buried deep in the vacuum. Further, interaction of a charged particle with the vacuum electrons through the quantum jitters gave the vacuum physical properties, such as vacuum polarization. A single charged particle attracted or repelled vacuum electrons, so that real particles had to be interpreted as a composite of a bare particle clothed with its effects in the distribution of vacuum electrons.

**Field Theory**

**Field Theory**

Max Planck had been able to analyse the cavity energies of electromagntic waves in black body radiation by reducing the continuum of wave motion into a discrete series of standing waves, each behaving like a harmonic oscillator. In this way he could treat the energy of a field as a network of thermally coupled harmonic oscillators, each of which could gain or lose discrete units of energy, h f, that were proportional to the frequency of that wave, f, the constant of proportionality being a fundamental constant, h. In a sense this was a brilliant use of statistical mechanics, not applied to particles, but to normal modes of vibration of extended waves. It was a work of amazing determination pursued by a Germanic scientist in an elaborate bookkeeping scheme; however, it was not really a quantum theory of the electromagnetic field itself.

Planck's work did contain a hint for how to develop a true quantum theory of electromagnetic waves interacting with electrons. His analysis leaned heavily on the mathematics developed by Joseph Fourier, which is called, Fourier Analysis. In Planck's analysis, the the standing waves used to calculate the energies formed an infinite, denumerable set. Fourier Integrals extend these methods to open, unlimited space, which involve a continuous distribution of frequencies. Planck's tricky analysis reduced the electro-magnetic field to a collection of harmonic oscillators, to which he assigned discrete energy levels. But then, this happens automatically, if we treat each harmonic oscillator quantum mechanically, something we have done already in a previous blog, "The 1-D Harmonic Oscillator in Quantum Mechanics". In this case, we have to treat the Fourier amplitudes, not as numbers selected for our pleasure, but as quantum mechanical operators through a procedure called second quantization. The mathematical theory of reducing extended fields to Fourier components, in which the amplitudes are quantum mechanical operators, is called Quantum Field Theory (QFT).

**The Bronx Guys**

I worked in QFT in my PH d thesis work at Case Western. My advisor, Joseph Weinberg was a boyhood friend of Julian Schwinger and Richard Feynman. Both were physicists from the Bronx who had won a Nobel Prize jointly with Sin-Itiro Tomonaga for their work in QFT in developing a theory of Quantum Electrodynamics (QED). I met and spoke with Julian Schwinger when he came at the invitation of Joseph Weinberg to give a seminar at Case Western University.

**Feynman Diagams**

Whereas the methods of Schwinger were rather sophisticated, they were also opaque. The clearest presentation of QED was by Richard Feynman, who invented a diagrammatic way of handling the various mathematical terms of a perturbation series expansion of the problem. Unfortunately, I never met Richard Feynman. For whatever political reason, Feynman diagrams were not viewed favorably by the senior staff of the Case Western physics department, so I had to do my calculations through a different way that I glommed together from the works of Freeman Dyson and others, which amounted to the same thing as Feynman diagrams.

Richard Feynman is another example of a physicist who achieved great success by following the dictum of "Shut up and calculate!" He was a calculating wizard, some people called him a magician because of the way he solved problems in physics, which was to think real hard a problem, then write out the solution as if by magic. The great calculational tool, he invented himself, was the Feynman diagram, in which an involved expression for a complex amplitude was represented by a diagram. He turned an abstruse, mathematical expression into an object, a Feynman diagram. The analyst was able to break down a problem into a series of Feynman diagrams that he could turn into mathematical expressions and evaluate at the end by converting those results into their mathematical equivalents. It was easier to think about the problem in this way, freeing the mind from carrying a series of complex, detailed expressions to carrying instead the graphical objects.

**Infinities and Renormalization**

**Infinities and Renormalization**

Consistently, infinities that kept showing up in the calculations were the big problem. The series of Feynman diagrams gave results that converged progressively onto the experimental results in terms of an expansion in powers of a term that representing the strength of the interaction, the coupling constant, provided that the coupling constant were smaller than unity. In the electromagnetic interaction, the coupling constant is small, about 1/137, so people hoped to be able to sum the series of Feynman terms to arrive at final answers; however, the problem of the very large number of possible vacuum interactions and quantum fluctuations gave a divergent series that gave infinities instead of finite values. To bypass the problems of infinities at this point, physicists developed a procedure called renormalization, which represented an off shoot of the "Shut up and calculate" philosophy.