Abstract:

CONFINED QUANTUM FIELD THEORY Mohammad Fassihi

Department of Mathematics Amirkabir University of Technology: 424, Hafex Ave.: P.O. Box 1 5875-4413

Tehran, Iran.

In this model state functions in space have compact support. It is proved that all algebraic structures of the quantum field theory are preserved. The momentum is a global conserved quantity, which makes it possible that the theory be compatible with the theory of relativity. This model answers some fundamental questions as locality. It is explicitly shown that

divergent terms in Feynman Rules for A.¢4 theory can be finite without renormalization. We

show also that this model can describes in more easier way phenomena in solid state as superconductivity and superfluidity. We also show that Feynman's conclusion of one photon experiment is a miss-interpretation. Based on Confined Quantum Field Theory a pre-super conducting state is constructed which solves many problems in solid state especially problem of super conductivity and superfluidity.

Introduction. The base of this theory is to give priority to the laws followed by the symmetry of the space, namely Noether theorem is over the quantum axioms. According to the Noether theorem (4) conserve quantities in physics are related to some fundamental symmetry. We develop this by stating each conserve quantity is breaking of some symmetry. Momentum is breaking the translational symmetry of the space and energy is the breaking the symmetry of the space in time. This is realised by choosing a bounded simply connected domain of the space and on this domain constructing the quantum system, which represent an elementary particle. Therefore the operator system and the state functions obeying the domain. And all state functions have compact supports, which are entirely on the domain, and operator acts on these state functions and are intrinsic part of the domain. To recognise energy as the breaking the symmetry of the space in time quantitatively establish a relation between energy and the metric of this domain. Generally topology of the domain represents the type of the particle and the metric the energy density. Therefore we get a relation between energy and the radius of confinement. Experimentally there are strong evidences confirming this relation. In high temperature for example in plasma physics particles acts as small balls (12) and the system can be treated in classical way. But in low temperature particles state functions occupies a large domain and we have overlaps of state functions and the system must be treated in quantum way. To show that it is possible to construct a quantum field system on a compact domain, we construct creation and annihilation operators acting on the Hilbert space of quadratic integrable functions with support on this domain. It is easy to see that commutation relations between these creation and annihilation operators are identical to those for unbounded domain (3) and therefore the algebraic structure is not changed by going from unbounded domain to the bounded.

Confined Quantum Field Theory is finite theory. To show this we take an explicit example in A.¢4 theory. The two points connected Green's function in this theory is the following:

0<2l(x1 , x2 ) = �(x1 - x, ) - � fd4y�(x1 - y)�(y -y)�(y -x2 ) + 0(A.' ) . 2 Q

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Usually terms like these are divergent in Feynman Rules Here we have advantage of the fact that

d4 !/i {.\-y) our integration domain is bounded. We show that the distribution t>.(x - y) = J�-e-2--2 (2Jr) p + m acts in a bounded way on the function 'f' (x) E L1 (Q) in the sense that

f� fd 4x-2e--2 'P(x) :S: Cll'¥(x)ll if 'f' (x) belongs to BMO. I d' ;px I (2JT) n p + m L, (nJ

Feynman's conclusion of one photon experiment is a miss-interpretation. In single photon expe1iment we observe an interference pattern which is the same as interference pattern due to the light passing through two tiny close slits. By this Feynman concluded (10) that we can never say from which slit the photon passes and therefore can never be localised. What is missing in his observation is that the photon we observe in the interference pattern is a secondary photon and is not the original. Here in fact we have no direct photon photon interaction, but photon by contacting electrons on the wall of the slit create an electronic waves which is non-local due to the electron-electron correlation. If energy quanta of this wave cannot be absorbed by the phonons and other assessable energy levels of the walls, it is reflected as a secondary photon. Since the electronic wave is non-local it is affected by both slits. Therefore interference pattern is affected by both slits even the single photon touches only one of them.

Confined Quantum Field Theory is a solution to the superconductivity and supcrfliudity. By foundation of the CQFT each quantum system possess a well-defined global conserved

momentum. When an electron moves in a periodic potential this momentum changes due to the integral of force exerted pointwise by the potential. The change of momentum changes the total energy of the s ystem and therefore the metric of the quantum system. Change of the metric changes the radius of the confinement Therefore domain of integration is function of the energy of the quantum system. For some radius of confinement the exchange of the energy with bulk is minimum. Since phonons possess discrete energy level, not all energies can be absorbed by the bulk. And for some radius of confinement if the exchange energy is lower than the minimum acceptable energy for the bulk, then there caru10t be any energy transfonn. In this case quantum system can move in the bulk without resistance. Here we have a single pre-superconductive electron. This can we demonstrate in one dimension ia the following way; suppose we are in one dimension, then our domain is a line segment and also suppose that our periodic potential is a sinus or cosine function, and charge density is uniformly distributed on the segment, then for the force on the segment we have;

F = dp = J psin(x)dx , where the p is the charge density, and we see that if the length of the dt " a+2n;r

segment ( [a,b]) = 2nJT , then J psin(x)dx = 0

, That means such a quantum system can move without resistance.

Application in solid state. When radius of confinement of an electron coincides with a number of the period of the potential the integral of the force exerted to the electron vanishes identically

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everywhere, and the electron can move without resistance unless they become disturbed by other electrons, phonons, impurity, defects or other elements that causes changes in the periodicity of the bulk. Therefore we have a discrete set of radius of confinement correspond to a discrete energy levels. If temperature is low and we have less impurity and defects a conducting electron brings more time in such states than the transitory states. The elements we mentioned above together with the junctions, like when we put to different metals together, or two semiconductors or Josephson junction, are the main actors in solid state. CQFT explain in a very simple way the phenomenon, which these actors create. Let take for example the thermoelectric effect. Thermoelectric effect is said to be due some potential barrier created in putting two different metal together. Many ask how can we have a potential barrier when the two metals are electrically neutral and it is a justified question. The fact is that we have no potential barrier at the junctions but only the change of periodicity. When --an electron moves in a metal or semiconductor most of the time is in the stable state or pre-superconductive state, which is when the radius of confinement is adjusted to the periodicity of the bulk. Then when this electron wants to pass the junction must go to another periodicity. And in that periodicity the radius of confinement is not adjusted to the periodicity. In order that the electron can pass the junction must exchange some energy. And if the second metal cannot accept these energy quanta, the electron reflects back. CQFT can explain also Josephson oscillation in a simple manner. We had described a presuperconducting electron. Superconducting state is when all presuperconductive electrons in same energy level move parallel and with the same distance from ·each other. Therefore the potential that an individual electron feels from the other is also periodic and therefore the force exerted by them also vanishes and all electrons can move collectively without resistance. In Josephson junction when an electron is reflect back from the junction is force again against the junction by the applied electric field. If these acts happen in a collective way the electron feels less resistance due the preserved periodicity. Therefore the collective reflection is more favourable and we experience an electric oscillation.

Boltzmann equation and transition to superconductivity. Boltzmann equation without external field has the form(l 2) fi f (p, x,t) + V.Vf (p, x, t) = f0011 , Where f (p, x,t) is the density of the particles with

momentum p at the time t in a little volume around the point x and V the velocity of the particle. Considering this equation for the conducting electrons. We will take the following facts into the consideration. If the electrons, which all belong to the pre-super-conducting energy levels, moving parallel to each other and uniformly are distributed and the potential is periodic, and we have no disturbing elements, there is no energy exchange between the individual electron and the rest of the system. Therefore in this case we can take fco11 = 0 . In other case f_.011 depends mainly on the deviation of the distribution from the constant. And we have;

.i_f(p,x, t) + V.Vf(p,x, t) = -Cf(p,x, t) , therefore we will find out that the solution is of the at

form f(p,x, t) = e-0'¥(p, x, t) where 'Y(p,x, t) is the solution to the

a - f (p, x,t) + V.Vf (p, x, t) = 0 at

It is easy to show that 'Y(p, x,t) which can be constructed with the sinus and cosine base is a bounded solution if the initial value is sufficiently regular and bounded in the sense that

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e-0'¥(p,x,t) � 0 as t � = . And therefore the system tends to the state of superconductivity, if the temperature is low.

Conclusion. Confined Quantum Field Theory not only solve most basic problems like locality and divergent problem and in this way takes away a lot of confusion existing in quantum theory. But also provide us with simple and powerful methods of solving problems in many branches in physics.

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