Confined Quantum Field Theory
Mohammad Fassihi
Department of Mathematics
Amirkabir University of Technology: 424, Hafex Ave.: P.O. Box 15875-4413
Tehran, Iran.
Abstract:
A consistent model for quantum field is constructed in which state functions in space have compact support. It is proved that all algebraic structures of the quantum field theory are preserved in this model. The momentum in this construction is a global conserved quantity. This 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 theory can be finite without renormalization.
And asymptotic freedom gets a physical explanation. 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. These states are those that theirs domain are a multiple of the periodicity of the bulk. It is shown that such a state do not exchange energy with the bulk with periodic potential.
If domains of some numbers of electrons are positioned in such a way that follow the periodicity of the bulk. These electrons do not exchange energy with neither the bulk nor with the pre-super conducting electrons. The stability character of such a collective states is compatible to the states 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, namly Noether theorem infront of the quantum axiums. According to the Noether theorem conserve quantities in physics are related to some fundamental symmetries. 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 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 and therefore the algebraic structure is not changed by going from unbounded domain to the bounded.
Confined Quantum Field Theory is finite theory. 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 explicit example in theory. The two points connected Green’s function in this theory is the following:
.
Usually terms like these are divergent in Feynman Rules. The reason is that people have no control over the terms like. Here we have advantage of the fact that our integration domain is bounded. We remind that the Green’s function is in fact a distribution. Since our domain is bounded if we show that this distribution is bounded by some constant function then the termsare under control. Here we need some assumption on the class of the function that this distribution acts on. Lets take the representation,
and the test function and without lose of generality put . We want to show that
for some constant . In this case the distribution is bounded upward by the distribution represented by the constant and the same is true for if we see as a limit for . We can divide integration in in two parts, small and big . For small the estimate is trivial. When the is big we can divide the integration’s domain according to . When grows the term oscillates as and if has bounded oscillation “belong to some BMO class of functions” the integral converges. In order to give the weakest possible condition we divide the into and . Here is the set in which oscillates faster than and its complement. Obviously and is decreasing. If then the integral converges. If there is a fixed number for which the second part is less than first part then the second part can simply be hidden in the first part. It is to say integration over finite . 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.
Experimentalists are more interested to work in the momentum space. This is due to the fact that especially in high-energy physics, most experiments 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 Feynmanns 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 example a free particle state function can be presented by;
Here is the characteristic function for the quantum domain and defined as
. And a normalization factor.
Lets calculate S-matrix
,
in the scalar neutral field with the interaction
to we get contributions. And during the calculation discover the similarities and the differences with the standard calculation.
,
We ignore factors like
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;
In the perturbation theory people often use the expression as function, which is true with some reservation. In order to become more familiar with the limits within it this application is valid, we calculate the following integral.
The first difficulty we are confronted with is the question of how we define the infinity. At list
one of the involved functions has no defined value at infinity. But the integral can gets a better definition if we let the integral limits to be for , for and then let .
We divide the space integration in two parts. And assume
By change of variables and , we have
As , if is continuous at the point , can be replaced by the constant .
Therefore we get;
.
The remaining integral is in principle an integral of sinus and cosines function over a fixed 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 and , which gives
As if . And (BMO is the class of function with bounded mean oscillation),
and the first part vanishes.
The second part is mainly contribution of the integration of the function at infinity, since
and if the function goes to zero strong enough as , the second term vanishes too.
Therefore in order to be able to represent the function by , the function on which this distribution acts must fulfil three conditions.
1- Continuity.
2- Bounded ness, and that .
3- The function strongly goes to zero as its argument goes to infinity.
4- 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 represents the class of domain on which our function is defined.
We can always define the function on a fix 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 , it is to say the space and time is extended to infinity. And since infinity is not well defined and unique, this domain is not either unique, however by fixing the way that we always must to go to the infinity, can fix such a domain. In other word fixing some , and keep in mind the way that we go to infinity. If we by name the class of domains we can construct a morfism, 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
(There represents the Green’s function and the differential operator.
There formally we have ),
(Here our function on which the function acts is . This function fulfils the above three conditions.)
There the constant depends on the domain and the way the function is defined on that domain.
There for
There stands for
Lets 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 and
Here is a ball with the radius of unity. Which if we assume the standard domain to be the ball with radius unity, gives us
By a dilatation in the space we cancel a factor to get
And finally we get
Comparing it with the standard calculation we observe partly the factor , which has to do with the way we normalize the incoming and outgoing state functions. And the factor , which in fact is the quantum volume. According to the CQFT radius of confinement 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 . Assuming that to be much less than the unity. Higher terms becomes smaller and smaller.
In addition we have the relation between the energy, and goes to zero as the energy goes to infinity. This is essential what we know as asymptotic freedom.
Feynman’s conclusion of one photon experiment is a miss-interpretation. In single photon experiment 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, 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.