Confined Quantum Field Theory

Transcription

1 Confined Quantum Field Theory Those challenges that we are confronted with in high-energy physics, and also those in lowest energy physics, namely superconductivity demands a stronger theory. This can be achieved by reducing and unifying the numbers of aioms in physics. One of the main aioms in physics is Noether theorem, connecting conserved quantities with the symmetries of the space. We take the step to have the breaking the symmetry of the space as the main aiom of physics. To demonstrate that breaking the symmetry is a stronger statement, we observe that the object below breaks the translational symmetry but not the rotational symmetry. Therefore we connect for eample momentum with the breaking the translational symmetry and the energy with breaking the symmetry of space in time. Then the first step in our physics is to break the symmetry of the space by considering a bounded domain and on this domain construct a quantum field. The construction of the quantum field is done by creation and annihilation operators. Creation and Annihilation Operator * We start with the creations operator ( f And the annihilations operator a ( f, a ( f = 0 in the global way. From the above we can define * a with the property a ( f = f for f H,

2 Pointwise Defined Operators a ( = f j( a( f j and j a ( = f j( a( f j. j f The Fourier transform (k of the function f ( H is follow; ik 3 f ( k = e f ( d, B The algebraic relations The commutations relations between creations an annihilation operators, using the definition is a straightforward calculation: [ a ( f, a ( ] g = ( f, g ± and [ ( f, a( g ] [ a ( f, a ( ] = 0 a. ± = g ± In the Fourier space these relations takes the form of [ a ˆ( k, aˆ ( k ] = $ ( k k. We observe that the algebra is identical with the algebra of the standard quantum field theory.

3 Feynman s conclusion of one photon eperiment is a missinterpretation. In single photon eperiment 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 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. Generally for waves interference we need some kind of boundary condition, which in this case is the electrons on the surface of the slits. Photons by contacting electrons on the wall of the slit create electronic waves, which is nonlocal 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 nonlocal 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 finite theory.

4 Since all the elementary particles are represented by a quantum system on a bounded domain, related space integration s are on a bounded domain. On the other hand since singularities in CQFT corresponds to infinite energy density, which is unphysical. All integration of a physical system is of regular functions on a bounded domain and therefore finite. Here we take an eplicit eample in theory. The two points connected Green s function in this theory is the following: ( G ( = $ ( $ ( $ ( $ ( + ( % d y y y y y. & Usually terms like these are divergent in Feynman Rules. The reason is that people have no control over the terms like d y ( y y. Here we have advantage of the fact that our integration domain is bounded. We remind that the Green s function ( y is in fact a distribution. Since our domain is bounded if we show that this distribution is bounded by some constant function then the terms d y ( y y are under control. Here we need some assumption on the class of the function that this distribution acts on. Lets take the representation, ip( y d p e ( y = ($ p + m generality put y = 0. We want to show that d p and the test function ( L ( and without lose of e ip $ (% $ d ( C ( p + m L ( for some constant C. In this case the distribution ( y is bounded upward by the distribution represented by the constant C and the same is true for ( y y if we see ( y y as a limit for ( y. We can divide integration in p in two parts, small p and big p. For small p the estimate is trivial. When the p is big we can divide the integration s domain according to n p n +. When ip n grows the term e oscillates as sin(n and if ( has bounded oscillation belong to some BMO class of functions the integral converges. In n n order to give the weakest possible condition we divide the into and. n n Here is the set in which ( oscillates faster than sin(n and its ( n + n ( n n complement. Obviously and m ( is decreasing. If sup n n { $ ( } m( C then the integral converges. If there is a fied number + n N for which the second part is less than first part then the second part can simply

5 be hidden in the first part. It is to say integration over finite p. And as we mentioned for the first part the estimate is trivial. Physically this condition is very weak and physical states functions enjoy much more regularity conditions than this. Application of Confined Quantum Field Theory in High Energy Physics. Eperimentalists are more interested to work in the momentum space. This is due to the fact that especially in high-energy physics, most eperiments are performed by targeting particles by another beams of particles. Here the spatial information for an individual particle in the beam is mostly irrelevant. However in CQFT interaction is due to the overlap of the state functions. Therefore in calculation of the cross section one should bring into the consideration the probability that two state functions have overlap. The quantum domain for a particle shrinks as its energy increases. This includes also the time of interaction. The estimate we presented which shows finiteness of the Feynman s terms is fundamental and abstract. Here we try to bring the CQFT closer to the more quantitative and practical calculations. Once again we mention the similarities and the differences between CQFT and standard QFT. The similarities are the algebraic structure, and the differences are mostly size and the topology the quantum domain. In CQFT we take a bounded manifold with nontrivial topology as the quantum domain. If we ignore for the moment the topology of the quantum domain, we can think of the form of the mathematics of the CQFT to be the same as the standard one but restricted to a bounded domain. For eample a free particle state function can be presented by; = C ( Ep[ i. p] Here ( is the characteristic function for the quantum domain and defined as for ( =. And C a normalization factor. 0 otherwise Lets calculate S-matri S fi =< k k, out k, k, in > 3,, in the scalar neutral field with the interaction g L 3 I = 3 And during the calculation discover the similarities and the differences with the standard calculation.

6 ( d d d 3 d ik ik + ik + ik S e e e e fi g k k k 3 k = & µ ' ' + $% µ & µ ' ' + µ $% 3 µ & µ µ ' ' + µ $% µ ( z ( z z ( z ( z 3, perm. of,,, 3, & µ ' ' + µ $% µ d z d z ( z We ignore factors like, i,... Here all spatial integration is over the bounded quantum domain, therefore we cancel the characteristic function (. Since this type of calculations function is frequently used, it is instructive to describe it little closer. function per definition is a distribution, which maps a function to a number. More closely we have; f ( ( a d = f ( a In the perturbation theory people often use the epression function, which is true with some reservation. In order to dke ik ( a as become more familiar with the limits within it this application is valid, we calculate the following integral. I = dk f ( e ik ( a d The first difficulty we are confronted with is the question of how we define the ik ( a infinity. At list one of the involved functions e has no defined value at infinity. And the value of such integrals mostly depends on the way that we go

7 to infinity. But the integral can gets a better definition if we let the integral limits &' &' n n to be $, % ( ( for k, $, % ( ( for and then let 0. We divide the space integration in two parts. And assume a = 0 I dk = f ( e ik d < n$ k $ I dk $ = f ( e ik d n k n% % % By change of variables k I = dk f ( e $ < n ik d and k k, we have As 0, if f ( is continuous at the point 0, can be replaced by the constant f (0. Therefore we get; I ik = f ( 0 dk e d. k < n The remaining integral is in principle an integral of sinus and cosines function over a fied volume and therefore can be replaced by a constant. We must emphasis that this constant much depends on the way that we go to infinity. To calculate the second integral we change the variables as which gives I = dk k $ % n n% f ( e d = dk f ( e d + ik ik % % $ n % > n% k k $ n n % % % dk and k k, f ( e % As 0 if f ( c. And f ( BMO (BMO is the class of function with bounded mean oscillation, k % ik dk f ( e d c d 0 and the first part vanishes. & & n & > $ n& n & > $ n& The second part is mainly contribution of the integration of the function at infinity, since ik d

8 and if the function f ( goes to zero strong enough as, the second term vanishes too. ik ( a Therefore in order to be able to represent the function by dke, the function on which this distribution acts must fulfil three conditions. - Continuity. - Bounded ness, and that f ( BMO. 3- The function strongly goes to zero as its argument goes to infinity. - And also we must be aware that the constant involved depend on the procedure that we take and the way we define the infinity. On the above construction n represents the class of domain on which our function is defined. We can always define the function on a fi domain, and if our calculation happens to be on the other domain, we can reach the function by some dilatation of the domain. In standard calculation the domain is R, it is to say the space and time is etended to infinity. And since infinity is not well defined and unique, this domain is not either unique, however by fiing the way that we always must to go to the infinity, can fi such a domain. In other word fiing some n, and keep in mind the way that we go to infinity. If we by name the class of domains we can construct a morphism, which maps the structure of the perturbation calculation from one domain to the other. Of course the topology of these domain is not trivial. But in the simplest case we can assume that all the domains are balls with the different radius, and the radius varies from zero to infinity. Further we can assume that these maps do not change the structure of Hamiltonian. In another word the operators of the Hamiltonian point wise acts in the same way in all domains. Therefore we get the same Green s functions, or propagators. The only change that we get is the scale of the domain, which reflects itself on the definition of the function. Lets first calculate d ik. e 0 k (There & µ $ ' ' % µ + µ [ ik ( z ] ep d k k µ + i$ [ ik ( z ] ep z = d k k µ + i$ G ( represents the Green s function and & µ D ( = $ ' ' + µ the differential operator. % µ There formally we have D ( G( z = ( z,

9 ik. (Here our function on which the function acts is ( e. This function fulfils the above three conditions. = d ik. e 0 k ( z. z = A eik (z 0 k There the constant A depends on the domain and the way the function is defined on that domain. There for ( S fi = A g $ % d z d There stands for ( k k k3 k z [ ik. z ik. z + ik. z + ik. z ] ( z z ep 3 perm. of,,, 3, [ ik. z ik. z + ik. z ik. z ] ( S fi A g d zd z ep 3 + $ = [ ik ( z z ] ep d k k µ + i$ [ iz ( k k k + iz ( k + k + k ] ep d k k µ + i ( 3 S fi = A g $ d zd z % & Lets R be the radius of confinement. As we mentioned before our definition of the function includes a defined way to go to infinity. Therefore we may rescale the system to get the space integration be over a ball with radius unity. This insures that in all cases the way of going to infinity becomes uniform. Then let S = RZ Z = RZ = $ ep A g ( R d z d z d k k µ + i % B Z and [ iz R( k k k + iz R( k + k + k ] ( 3 fi Here B is a ball with the radius of unity. Which if we assume the standard domain to be the ball with radius unity, gives us & R( k k k & R( k + k + k [ ] [ ] S fi = A g ( R $ d k k µ + i% ( 3 By a dilatation in the k space we cancel a factor R to get ( & ( k k k & ( k + k3 + k = A g R $ d k k µ + i% S fi And finally we get

10 S fi ( = A g R ( ak + ak ak 3 ak Comparing it with the standard calculation we observe partly the factor A, which has to do with the way we normalize the incoming and outgoing state functions. And the factor R, which in fact is the quantum volume. According to the CQFT radius of confinement R is function of energy and decreases with increased energy. In perturbation calculations higher order terms involves higher number of space integration. Each space integration gives us the factor R. Assuming that R to be much less than the unity. Higher terms becomes smaller and smaller. In addition we have the relation between R and the energy density, and R goes to zero as the energy goes to infinity. This is essential what we know as asymptotic freedom. k + k Confined Quantum Field Theory is a solution to the superconductivity and superfliudity. 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 eerted pointwise by the potential. The change of momentum changes the total energy of the system 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 echange 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 echange energy is lower than the minimum acceptable energy for the bulk, then there cannot be any energy transform. 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 in 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;

11 b dp F = = sin( d, where the is the charge density, and we see that if the dt a length of the segment ( [ b] a+ n a, = n, then sin( d = 0, That means such a quantum system can move without resistance. a 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 eerted to the electron vanishes identically 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 eplain in a very simple way the phenomenon, which these actors create. Let take for eample 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

12 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 echange some energy. And if the second metal cannot accept these energy quanta, the electron reflects back. CQFT can eplain 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 eerted 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 eperience an electric oscillation. The picture demonstrates a junction, and the conducting electrons passing the junction.

13 Criticism of BCS Theory BCS theory assumes that when the temperature is low enough two electrons are coupled to build a par, which is partly true. Since in reality in the superconducting state conducting electrons move in a collective way and therefore the electrons always have the same neighbour. More this theory is not capable to eplain, and there are many question unanswered in this theory. BCS theory has the following question to answer; - Spin ½ and spin are topological concept. And to go over topological barrier is not physically proven and is not as simple as / +/ =. - Electrical resistance in solids depends on electrical charge and not on the type of the particle. Even if a particle assumed topologically be a Boson, it does not means not to eperience resistance. 3- Superconductivity and superfluity share the same basic phenomena. Therefore to describe superfluity we must assume that two helium s build up an object, which is Bosonic, which is both practically and theoretically impossible to prove. Boltzmann equation and transition to superconductivity. Boltzmann equation without eternal field has the form t Where f ( p, + V. f ( p, = f coll, f ( p, is the density of the particles with momentum p at the time t in a little volume around the point 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 echange between the individual electron and the rest of the system. Therefore in this case we can take f coll = 0. In other case f coll depends mainly on the following parameters; Deviation of the electrons distribution from a constant. In such a situation electrons are scattered by each other s repulsive potential. Impurities cause the

14 potential deviation from the periodicity and therefore scatter the electrons. The same is the effects of defects in crystal. The scattering of the electrons due to the impurity and defects strongly depends on the temperature. If the temperature is low most of the pre-super-conducting electrons have low energy and therefore theirs quantum domain cover a large area, which can be much larger than the area of the impurity or the defects. In collision when the electron passes the impurity or defects mostly covers all the impurity or defects and the integral of the effects mostly vanishes. And in all cases when the disturbing energy is lower than acceptable energy level for the crystal disturbance can be neglected. Considering that these disturbing can have linear effect on collision term, we can write the Boltzmann equation for the highest energy pre-super-conducting electron in this way; t f ( p, + V. f ( p, = Cf ( p, + f imp + f def Here we take the electrons in the highest pre-super-conducting level due to the fact that they are more affected by the disturbances and if we have transition to super-conductivity in this level we have also for the lower levels. If we take f ( p, to be the deviation of the density from a constant. If the temperature is low we can forget the terms f imp + f def and look for the solution for the rest of the equation. We will find out that the solution is of the form f ( p, = e Ct ( p, where ( p, is the solution to the t f ( p, + V. f ( p, = 0 It is easy to show that ( p, 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 e Ct ( p, 0 as t. And therefore the system tends to the state of superconductivity, if the temperature is low.

15 Integrity of the particles. In CQFT integrity of elementary particles follows from the relation between energy density and the metric of the quantum system. Since increase in energy density causes more confinement, then any eternal energy source directed to divide a system causes more confinement and therefore creates the opposite effect. 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 eisting in quantum theory. But also provide us with simple methods of solving problems in many branches in physics. References. D. F. Walls, G.J. Milburn Quantum Optics, Springer-Verlag 99. Steven Weinberg The Quantum Theory of Fields Cambridge University press G. Scharf, Finite Quantum Electrodynamics, Springer-Verlag 995. E. Neother, Nachr. Akad. Wiss. Goettingen Math. Phys. Kl. p. 35 (98 5. Siegmund Brandt, Hans Dieter Dahmen Quantum Optics, Springer-Verlag H. Kamerlingh Onnes. (9 7. Neil W. Ashcroft, N. David Mermin Solid State Physics, Holt-Saunders International Editions C. W. Kilmister, Hamiltonian Dynamics. Amer. Elsevier, New York, Pierre Ramond, Field Theory FIP. Lecture Note Series, Addison-Wesley Publishing Company, Feynman, QED, PRINCETON UNIVERSITY PRESS (985..P.V.E. McClintock, Low-Temperature Physics, Blackie(99