Relationship of physics to mathematics

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relationship of physics to mathematics

The connection between mathematics and the description of the universe goes far The dominating idea in this application of mathematics to physics is that the . Physics > History and Philosophy of Physics The Mysterious Connection Between Physics and Mathematics", pp. , (Springer, ). The essay is in the form of a dialogue between the two authors. We take John Wheeler's idea of “It from Bit” as an essential clue and we rework the structure of .

A physicist has won the Fields Medal, one of the most prestigious accolades in mathematics. And a mathematician, Maxim Kontsevich, has won the new Breakthrough Prizes in both mathematics and physics.

One can attend seminar talks about quantum field theory, black holes, and string theory in both math and physics departments. Sincethe annual String Math conference has brought mathematicians and physicists together to work on the intersection of their fields in string theory and quantum field theory.

String theory is perhaps the best recent example of the interplay between mathematics and physics, for reasons that eventually bring us back to Einstein and the question of gravity. String theory is a theoretical framework in which those pointlike particles Dirac was describing become one-dimensional objects called strings.

Most humans will tell you that we perceive the universe as having three spatial dimensions and one dimension of time. But string theory naturally lives in 10 dimensions. Inas the number of physicists working on string theory ballooned, a group of researchers including Edward Witten, the physicist who was later awarded a Fields Medal, discovered that the extra six dimensions of string theory needed to be part of a space known as a Calabi-Yau manifold.

When mathematicians joined the fray to try to figure out what structures these manifolds could have, physicists were hoping for just a few candidates.

Instead, they found boatloads of Calabi-Yaus. Mathematicians still have not finished classifying them. As mathematicians and physicists studied these spaces, they discovered an interesting duality between Calabi-Yau manifolds. Two manifolds that seem completely different can end up describing the same physics. This idea, called mirror symmetry, has blossomed in mathematics, leading to entire new research avenues.

The framework of string theory has almost become a playground for mathematicians, yielding countless new avenues of exploration. Mina Aganagic, a theoretical physicist at the University of California, Berkeley, believes string theory and related topics will continue to provide these connections between physics and math. Mathematicians and their focus on detailed rigorous proofs bring one point of view to the field, and physicists, with their tendency to prioritize intuitive understanding, bring another.

There is seemingly no end to the places where a well-placed set of tools for making calculations could help physicists, or where a probing question from physics could inspire mathematicians to create entirely new mathematical objects or theories. Possibly, the two subjects will ultimately unify, every branch of pure mathematics then having its physical application, its importance in physics being proportional to its interest in mathematics.

The unreasonable relationship between mathematics and physics

At present we are, of course, very far from this stage, even with regard to some of the most elementary questions.

For example, only four-dimensional space is of importance in physics, while spaces with other numbers of dimensions are of about equal interest in mathematics.

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It may well be, however, that this discrepancy is due to the incompleteness of present-day knowledge, and that future developments will show four-dimensional space to be of far greater mathematical interest than all the others.

The trend of mathematics and physics towards unification provides the physicist with a powerful new method of research into the foundations of his subject, a method which has not yet been applied successfully, but which I feel confident will prove its value in the future. The method is to begin by choosing that branch of mathematics which one thinks will form the basis of the new theory. One should be influenced very much in this choice by considerations of mathematical beauty.

It would probably be a good thing also to give a preference to those branches of mathematics that have an interesting group of transformations underlying them, since transformations play an important role in modern physical theory, both relativity and quantum theory seeming to show that transformations are of more fundamental importance than equations.

Having decided on the branch of mathematics, one should proceed to develop it along suitable lines, at the same time looking for that way in which it appears to lend itself naturally to physical interpretation.

relationship of physics to mathematics

This method was used by Jordan in an attempt to get an improved quantum theory on the basis of an algebra with non-associative multiplication. The attempt was not successful, as one would rather expect, if one considers that non-associative algebra is not a specially beautiful branch of mathematics, and is not connected with an interesting transformation theory.

I would suggest, as a more hopeful-looking idea for getting an improved quantum theory, that one take as basis the theory of functions of a complex variable. This branch of mathematics is of exceptional beauty, and further, the group of transformations in the complex plane, is the same as the Lorentz group governing the space-time of restricted relativity.

One is thus led to suspect the existence of some deep-lying connection between the theory of functions of a complex variable and the space-time of restricted relativity, the working out of which will be a difficult task for the future. Let us now discuss the extent of the mathematical quality in Nature. According to the mechanistic scheme of physics or to its relativistic modification, one needs for the complete description of the universe not merely a complete system of equations of motion, but also a complete set of initial conditions, and it is only to the former of these that mathematical theories apply.

The latter are considered to be not amenable to theoretical treatment and to be determinable only from observation. The enormous complexity of the universe is ascribed to an enormous complexity in the initial conditions, which removes them beyond the range of mathematical discussion.

relationship of physics to mathematics

I find this position very unsatisfactory philosophically, as it goes against all ideas of the unity of Nature. Anyhow, if it is only to a part of the description of the universe that mathematical theory applies, this part ought certainly to be sharply distinguished from the remainder. But in fact there does not seem to be any natural place in which to draw the line. Are such things as the properties of the elementary particles of physics, their masses and the numerical coefficients occurring in their laws of force, subject to mathematical theory?

According to the narrow mechanistic view, they should be counted as initial conditions and outside mathematical theory. However, since the elementary particles all belong to one or other of a number of definite types, the members of one type being all exactly similar, they must be governed by mathematical law to some extent, and most physicists now consider it to be quite a large extent. For example, Eddington has been building up a theory to account for the masses.

But even if one supposed all the properties of the elementary particles to be determinable by theory, one would still not know where to draw the line, as one would be faced by the next question - Are the relative abundances of the various chemical elements determinable by theory? One would pass gradually from atomic to astronomic questions.

relationship of physics to mathematics

This unsatisfactory situation gets changed for the worse by the new quantum mechanics. In spite of the great analogy between quantum mechanics and the older mechanics with regard to their mathematical formalisms, they differ drastically with regard to the nature of their physical consequences.

According to the older mechanics, the result of any observation is determinate and can be calculated theoretically from given initial conditions; but with quantum mechanics there is usually an indeterminacy in the result of an observation, connected with the possibility of occurrence of a quantum jump, and the most that can be calculated theoretically is the probability of any particular result being obtained.

The question, which particular result will be obtained in some particular case, lies outside the theory. This must not be attributed to an incompleteness of the theory, but is essential for the application of a formalism of the kind used by quantum mechanics.

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Thus according to quantum mechanics we need, for a complete description of the universe, not only the laws of motion and the initial conditions, but also information about which quantum jump occurs in each case when a quantum jump does occur. The latter information must be included, together with the initial conditions, in that part of the description of the universe outside mathematical theory. The increase thus arising in the non-mathematical part of the description of the universe provides a philosophical objection to quantum mechanics, and is, I believe, the underlying reason why some physicists still find it difficult to accept this mechanics.

Quantum mechanics should not be abandoned, however, firstly, because of its very widespread and detailed agreement with experiment, and secondly, because the indeterminacy it introduces into the results of observations is of a kind which is philosophically satisfying, being readily ascribable to an inescapable crudeness in the means of observation available for small-scale experiments. The objection does show, all the same, that the foundations of physics are still far from their final form.

We come now to the third great development of physical science of the present century - the new cosmology. This will probably turn out to be philosophically even more revolutionary than relativity or the quantum theory, although at present one can hardly realize its full implications. The starting-point is the observed red-shift in the spectra of distance heavenly bodies, indicating that they are receding from us with velocities proportional to their distances.

If we go backwards into the past we come to a time, about 2 x years ago, when all the matter in the universe was concentrated in a very small volume.

The unreasonable relationship between mathematics and physics |

It seems as though something like an explosion then took place, the fragments of which we now observe still scattering outwards. With this kind of cosmological picture one is led to suppose that there was a beginning of time, and that it is meaningless to inquire into what happened before then.

One can get a rough idea of the geometrical relationships this involves by imagining the present to be the surface of a sphere, going into the past to be going in towards the centre of the sphere, and going into the future to be going outwards.

There is then no limit to how far one may go into the future, but there is a limit to how far one can go into the past, corresponding to when one has reached the centre of the sphere. The beginning of time provides a natural origin from which to measure the time of any event. The result is usually called the epoch of that event. Thus the present epoch is 2 x years. Let us now return to dynamical questions. With the new cosmology the universe must have been started off in some very simple way.

What, then, becomes of the initial conditions required by dynamical theory? Plainly there cannot be any, or they must be trivial. We are left in a situation which would be untenable with the old mechanics. If the universe were simply the motion which follows from a given scheme of equations of motion with trivial initial conditions, it could not contain the complexity we observe. Quantum mechanics provides an escape from the difficulty. It enables us to ascribe the complexity to the quantum jumps, lying outside the scheme of equations of motion.

relationship of physics to mathematics

The quantum jumps now form the uncalculable part of natural phenomena, to replace the initial conditions of the old mechanistic view.

One further point in connection with the new cosmology is worthy of note. At the beginning of time the laws of Nature were probably very different from what they are now. Thus we should consider the laws of Nature as continually changing with the epoch, instead of as holding uniformly throughout space-time. This idea was first put forward by Milne, who worked it out on the assumptions that the universe at a given epoch is roughly everywhere uniform and spherically symmetrical.

I find these assumptions not very satisfying, because the local departures from uniformity are so great and are of such essential importance for our world of life that it seems unlikely there should be a principle of uniformity overlying them.

relationship of physics to mathematics

Further, as we already have the laws of Nature depending on the epoch, we should expect them also to depend on position in space, in order to preserve the beautiful idea of the theory of relativity there is fundamental similarity between space and time.