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The theory of Klein-Kaluza

The scientific approach 07.10.2016 at 12:53

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, the Space of Klein-Kaluza

the day before his death, Einstein asked that he filed his latest calculations on the unified field theory. For thirty years he fruitlessly attempted to modify General relativity to include the electromagnetic force. One of the most promising options came to Einstein in 1919, at the beginning of his quest, while he was sorting mail. The idea came to his mind, not directly, but through a letter from one beggar mathematician named Theodor Kaluza.

In the letter Einstein found the proposition of how to combine the electromagnetic force with gravity. This theory was one little oddity. Einstein wrote in response: "the Idea of creating a unified theory] by means of five-dimensional cylindrical world never crossed my mind..." five-dimensional cylinder? But who could such a thing come to mind? No one knows how Kaluza thought of this before, but Einstein in the same letter he added: "I extremely cute your idea". Now we understand that Kaluza ahead of time, but got greedy measurements.

the General theory of relativity describes how matter affects space through the metric, whose components are g – factors report exactly how to measure the distance between adjacent points on the basis of the difference of their coordinates. The number of g-factors depend on the number of dimensions of space. For example, in three-dimensional space there are six. In the distance is flat (the difference between the coordinates of x)2 + (difference between coordinates)2 + (the difference between the coordinates z)2, i.e. gxx, gyy and gzz are all equal to 1 and the factors corresponding to the cross – gxy, gxz, and gyz are all equal to zero, and they are not in the equation. In four-dimensional non-Euclidean space of General relativity goes ten independent g-factors (taking into account the equality of the type gxy = gyx), all described by Einstein's equations. Kaluza first realized this: if you take five measurements, there will be another g-factors corresponding to an additional external dimension.

Further Kaluza wondered if formally expand einsteinova field, up to five measurements, which equations are obtained for the additional g-factors? The answer is stunning: out of the Maxwell equations for the electromagnetic field! Starting with the fifth dimension of electromagnetism suddenly arises in the theory of gravity. Einstein wrote: "the Formal unity of your theory is striking".

of Course, the interpretation of metrics additional measurements as a physical electromagnetic field requires some fuss with the theory. And there, by the way, with the smallest oddity – an extra dimension? Kaluza argued that it is certainly in length, and more to the point is so small that we would had not noticed him, even if they swarmed inside. Moreover Kaluza said that the new measurement has new topology: it is a straight line – circumference, i.e., it closes on itself, coagulates (and therefore, unlike the final straight, all has not). Imagine, Patowary with zero width in the form of a simple line. A new dimension in the Kaluza across the street will turn into a circle, proizvoditsa of Fifth Avenue. Of course, cross streets occur at intervals of a quarter, but there is an extra dimension at each point along the Avenue. So if you add line a new dimension, it will not acquire circles, and turn the cylinder like a garden hose. Only very thin.

In fact, Kaluza argued that gravity and electromagnetism are really components of the same, but look different because we are seeing some average tiny imperceptible movement of the fourth spatial dimension. Einstein doubted the theory of Kaluza, however, a little later still he changed his mind and in 1921 helped Kaluza to publish his theory.

In 1926, Oscar Klein, assistant Professor at the University of Michigan, regardless of Kaluza suggested the same theory, but with some improvements. One of them is the realization that this theory leads to the correct equations of motion for a particle, if in this mysterious fifth dimension, the particle has definite momentum values. These "allowed" values are multiples of a certain minimum momentum. If we assume, as did Kaluza, fifth dimension is closed on itself, it is possible to apply quantum theory to calculate the minimum momentum possible value of "length" that collapsed the fifth dimension. If suddenly it turned out that the dimension that is observable, macroscopic size, the theory would have been under threat because of the dimension we can not observe. But it turned out the size of 10-30 cm. No problem. The dimension is hidden from the eyes be healthy.

the Theory of Klein-Kaluza hinted at a formal link between the theories, but not to the structure, which immediately gave something completely new. The next few years, physicists have searched for other predictions, which could give this theory, approximately in the same key, in which Kline argued about the dimensions a new dimension. They managed to find new arguments, which seems to be implied that it can help to predict the ratio of the mass of the electron and its charge. However, the result of the prediction is strongly at odds with reality. Somewhere halfway between this difficulty and the strange prophecy of the fifth dimension physics lost interest in the new theory. Einstein the last time came back to her in 1938.

Kaluza, who died a year before Einstein, and almost no progress on. But something with his fledgling theory him big trouble. When he wrote to Einstein, he was 34 and he was already ten years have supported a family on the salary of assistant Professor (the approximate equivalent of assistant Professor) in königsberg. This is the the salary is best described in terms dear to his heart of mathematics: for every semester he received 5 times for x time DM (or, strictly speaking, gold marks), where x was equal to the number of students in his class, and the number of hours of lecture weekly. The result was about 100 marks a year. In 1926, Einstein called such living conditions "schwierig", which roughly means "only dogs can live like that." With povstanska Kaluza in 1929 finally received a professorship at the University of Kiel. He moved to göttingen in 1935, where he became a full Professor. There he lived for another nineteen years allotted to him. However, until the 1970s, the possibility of new measurements never seriously considered one.

the Birth of the strings

Who knows when there will rush of inspiration? Still impossible to know where it will lead. The history of string theory starts at the top of 750-foot mountain in the Mediterranean. Town called Erice in Sicily, slow, hot, the streets are narrow, and dressed in the ancient stone. Erice Erice was when the Earth still roamed the Thales. Now, the town is famous primarily for his "Centro Ettore Majorana" – a cultural and scientific center, which for decades held a summer school about a week length. School "Ettore Majorana" – a gathering of senior students and Junior members of the faculties, where they meet with leading scientists from different areas and attend lectures on the most advanced topics of science.

in the Summer of 1967 one of such advanced topics were the approach to the elementary particle theory called "the theory of S-matrices". Gabriele Veneziano, an Italian graduate of the Weizmann Institute in Israel, was in the audience and listened to his intellectual hero, Murray Gell-Mann. Gell-Mann will soon receive for his discovery of quarks the Nobel prize – they were at that time regarded the internal components of the family of elementary particles called hadrons (in the same family includes the proton and neutron). The inspiration Veneziano gained in the lecture in a few years will encourage him to create started on string theory. The theme of the lecture then the Gell-Mann was the pattern of the mathematical structure of the S-matrix.

S-matrix invented by Heisenberg, for the first time in 1937, used John Wheeler, and the Golden age was in the 1960s, and provided it with a physicist from Berkeley Jeffrey Chu. The letter S is denoted by "scattering" [scattering] as the main method of the study of elementary particles by physicists is that physics acceleration of particles to high speeds and energies, and then uspevayut them at each other and see what the pieces will fly in all directions. About how to study the structure of the car by arranging the car accident.

small accidents could tear something boring, like a bumper, but at race pace eyes the gaze of the observer will be flying even the most tightly wound into the passenger seat bolts and nuts. But there is one big difference. In experimental physics, punched scale "Chevy" "Ford", you can get the parts from Jaguar. Unlike cars, elementary particles can turn into each other.

When Wheeler developed the scattering matrix has already been gathered – and continued to accumulate a considerable body of experimental data, however, the successful quantum theory of creation and disappearance of elementary particles did not exist even in terms of electrodynamics. S-matrix was a black box, which could be something to put – definition of colliding particles, their momenta, etc. – and to output the same data, but for a newly formed particles.

For constructing a scattering matrix, i.e., the innards of the black box, generally speaking, required theory of interaction of particles. But even without theory is something about the S-matrix it is possible to say – based only on natural symmetries and General principles like consistency with the theory of relativity. Salt of the S-matrix approach was to find out how far you can go on some of these principles.

50-ies and 60-ies of the last century this approach was almost a craze. In his lecture in Erice Gell-Mann talked about some of the amazing patterns, called dualities, which can be observed in the collision of hadrons. Veneziano wondered if there were any such laws in a more General case. A year and a half he realized that all mathematical properties of the scattering matrix, which he considered inherent in one simple mathematical functions – Euler's beta function.

the theory of the Veneziano (dual Veneziano model) were amazing discovery. Why would a potentially complex scattering matrix to accept such a simple graceful form? But that was the first mathematical miracle in a series of many, which then will regularly appear in string theory – just such beautiful results convinced Schwarz that he is not wasting his life on string theory.

the Result obtained Veneziano, physicists seemed so elegant that inspired them to absolutely not S-matrix question: how is the process of collision of particles, which is obtained by the scattering matrix? What is the black box inside? If we could figure that out would clarify the internal structure of the colliding hadrons and the interaction, called the strong one that controls them.

In 1970, Eitiro Nambu from the University of Chicago, Holger Nielsen of the Niels Bohr Institute, and Leonard Susskind from Yeshiva University, answered the question: you need to simulate elementary particles not as points, but as tiny vibrating strings.

We are opening the theory or invent? Physics – children, wandering at dusk through the Park with lanterns in search of truth, or children with blocks, building towers until they crumble? Or, actually, – both? Then what kind of this duality – like the one mentioned by Gell-Mann, or as that is, waves and particles?

There are also less pleasant synonyms for the verb "to invent" and "open". For example, "to cook" or "stumble on". The original string theory, called bosonic string theory – was definitely "cooking". Lacked naturalness, it is full of incredible properties, and it is clearly gathered in a bunch, just to play inspiration, visit Veneziano. But Nambu with colleagues on something and stumbled. They discovered the string theory almost in the same sense that Planck time quantum. Both came upon the idea of energy levels it is possible to represent quantitatively, and the particles can be represented as strings; in both cases, neither the true value nor the breadth of these ideas were not understood, and the formation of a meaningful theory required years. Both came across what could be a new law of nature or mathematical grimace. And only years of effort can determine which is which. In the case of quantum theory it took 25 years – from Planck to Heisenberg and Schrodinger. String theory has already passed this milestone.

book Excerpt Leonard Mlodinow "Euclidean window. The story of geometry from parallel lines to hyperspace"

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