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.
4
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
 4 theory.
The two points connected Green’s function in this theory is the following:
G ( 2 ) ( x1 , x 2 )  ( x1  x 2 ) 

d
2

4
y( x1  y )( y  y )( y  x 2 )  (2 ) .
Usually terms like these are divergent in Feynman Rules. The reason is that people have no control over the terms
like
d
4
y( y  y ) . Here we have advantage of the fact that our integration domain is bounded. We remind that
the Green’s function  ( x  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
4
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,
( x  y )  
d 4 p e ip( x  y )
(2 ) 4 p 2  m 2
and the test function ( x)  L1 ()
and without lose of
generality put y  0 . We want to show that
d4p
e ipx
4
d
x
 (2 ) 4  p 2  m 2 ( x)  C ( x)
L1 (  )
for some constant C . In this case the distribution  ( x  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  ( x  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
and if  (x)
n  p  n  1 . When n grows the term e ipx oscillates as sin( nx)
has bounded oscillation “belong to some BMO class of functions” the integral converges. In order
1n and  2n . Here 1n is the set in which
 (x) oscillates faster than sin( nx) and  2n its complement. Obviously 1( n1)  1n  1( n1) and
1
m(1n ) is decreasing. If sup 1n  ( x)m(1n )  C 2 then the integral converges. If there is a fixed
n
number N 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 p . And as we mentioned for the first part the estimate is trivial. Physically this
to give the weakest possible condition we divide the  into
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 highenergy 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;
  C ( x) Exp ix. p
Here  (x ) is the characteristic function for the quantum domain 
 ( x) 
1 forx  
. And C a normalization factor.
0 otherwise
and defined as
Lets calculate S-matrix
S fi  k3 , k 4 , out k1 , k 2 , in  ,
in the scalar neutral field with the interaction
LI 
g 3

3!
2
to g we get contributions. And during the calculation discover the similarities and the differences with the
standard calculation.
S fi
( 2)
 g2


d 4 x1 d 4 x2 d 4 x3 d 4 x 4 ik1.x1 ik2.x2 ik3.x3 ik4.x4
e
e
e
e
k 01 k 0 2 k 0 3 k 0 4
     2 
  2

     2 
  3

     2 
  4


perm.of , x1 , x2 , x3 , x4

     2 
x
 

1
d 4 z d 4 z ( x  z )
1
2
1
1

 ( x  z )  ( z  z )  ( x  z ) ( x  z ) ,
2
1
1
2
3
We ignore factors like 2 , i,....
2
4
2
Here all spatial integration is over the bounded quantum domain, therefore we cancel the characteristic function
 (x) .
Since this type of calculations  function is frequently used, it is instructive to describe it little closer.
per definition is a distribution, which maps a function to a number. More closely we have;
 f ( x) ( x  a)dx 

function
f (a)
In the perturbation theory people often use the expression
 dke
 ik ( x  a )
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.
I   dk  f ( x)e  ik ( x  a ) dx
The first difficulty we are confronted with is the question of how we define the infinity. At list
one of the involved functions e
 ik ( x  a )
has no defined value at infinity. But the integral can gets a better definition if
1 1 
 n n
,  for k , 
,
for x and then let   0 .
  
   
we let the integral limits to be 
We divide the space integration in two parts. And assume
I1 

k
I2 
 f ( x)e
dk
 ikx
dx
x n
1


k
dk
1
n


 f ( x )e
 ikx
dx
 x  n
By change of variables
x  x and k 
1

k , we have
a0
I1 
 dk  f (x)e
k 1
 ikx
dx
x n
As   0 , if f (x ) is continuous at the point
Therefore we get;
I 1  f (0)  dk
k 1
e
 ikx
0 , can be replaced by the constant f (0) .
dx .
x n
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 x 
I2 
 dk 
k

2
 0
As
 dk
k
1
n  x  n 2
1
x
f ( )e ikx dx 

 dk 
k
1

n   x  n 2
1

x and k  k , which gives
x
f ( )e ikx dx 

2
 dk 
k
1
2
n x  n 
x
f ( )e ikx dx

x
if f ( )  c . And f ( x)  BMO (BMO is the class of function with bounded mean oscillation),


n   x  n 2
x
f ( )e ikx dx  c

 dx
 0 and the first part vanishes.
n   x  n 2
2
The second part is mainly contribution of the integration of the function at infinity, since
x

  and if the function f (x ) goes to zero strong enough as x   , the second term vanishes too.
Therefore in order to be able to represent the

function by
 dke
 ik ( x  a )
, the function on which this distribution
acts must fulfil three conditions.
1- Continuity.
2- Bounded ness, and that f ( x)  BMO .
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 n 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.
4
In standard calculation the domain is R , 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 n , 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

d 4x
k

(There
0
e
ik .x 

  


G ( x, z )  d 4 k 

 2


d 4k
exp  ik ( x  z )
k  2   2  i
exp  ik ( x  z )


represents the Green’s function and D( x)   
2
2

k     i


 2

the differential operator.
There formally we have D ( x)G ( x, z )   ( x  z ) ),
(Here our function on which the


d 4x

e
k0
ik .x

function acts is
 ( x  z)
 ( x)e ik. x . This function fulfils the above three conditions.)
Ae
 ik .z
k0
 (z )
There the constant A depends on the domain and the way the

function is defined on that domain.
There for
S fi
( 2)
 A 4 g 2   d 4 z1d 4 z 2 exp  ik1 .z1  ik 2 .z1  ik 3 .z 2  ik 4 .z 2 ( z1  z 2 )

There

 stands for k1 k 2 k 3 k 4
0
0
0

1
0 2

perm.of , x1 , x2 , x3 , x4
S fi
( 2)
 A4 g 2  d 4 z1d 4 z 2 exp  ik1 .z1  ik 2 .z1  ik 3 .z 2  ik 4 .z 2 

S fi
( 2)
 A 4 g 2   d 4 z1 d 4 z 2


d 4k

d 4k
exp  ik ( z1  z 2 )
k  2   2  i
exp  iz1 (k   k1  k 2 )  iz 2 (k   k 3  k 4 )
k  2   2  i
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 Z 1  RZ 1 and
Z 2  RZ 2
S fi
( 2)

 A 4 g 2 ( R 4 ) 2   d 4 z1d 4 z 2
B

d 4k
exp  iz1 R(k   k1  k 2 )  iz 2 R(k   k 3  k 4 )
k  2   2  i
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
S fi
( 2)

 A4 g 2 (R 4 ) 2  d 4 k 
By a dilatation in the
 R(k   k1  k 2 ) R(k   k 3  k 4 )
k  2   2  i
k space we cancel a factor R 4 to get
S fi
( 2)

 A4 g 2 R 4 d 4 k 
 (k   k1  k 2 ) (k   k 3  k 4 )
k  2   2  i
And finally we get
S fi
( 2)
 A 4 g 2 R 4 (ak1  ak 2  ak 3  ak 4 )
1
k1  k 2
A 4 , which has to do with the way we
Comparing it with the standard calculation we observe partly the factor
4
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
4
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 the energy, and R 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.
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 exerted 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 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
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;
b
F
dp
  sin( x)dx , where the  is the charge density, and we see that if the length of the segment (
dt a
a  2 n
a, b) = 2n , then   sin( x)dx  0
a
, 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 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 presuperconductive 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.
Universal application to both superconductivity and superfluity. The model we give here is not
restricted to a specific type of particles. The only condition demanded is that the quantum system has a radius of
confinement, which is function of energy and moving in a periodic potential. Therefore we can apply it both to the
electrons in superconductivity and helium in superfluity.
Application in statistical physics. According to the CQFT each particle occupies a bounded domain in the
space, with the radius of confinement depending on the energy. Then the probability that the state functions have
overlaps or not depends mainly on the energy of the system. Therefore we can construct a universal statistic, which
covers both Fermi-Dirac and the classical. London has estimated the energy of confinement in the crystals to
E confinement  C
R2
where
C be a constant and R radius of confinement. This estimate is in accordance with the
general idea of the CQFT and can be used in our calculations. Then if we have a cube with the size d  d  d we can
accommodate one particles with the energy C
with the energy C
4
16
3
, 8 Particles with the energy C 2 and generally i Particles
2
d
d
(2i) 2
. Then we can calculate the probability function to be
d2
W ( N1 ,.....N k ,.......)  CN!
i
(i 3 )!
3
with the restrictions N i  i ,
2 3
( N i !) (i  N i )!
N
i
 N and
i
(2i) 2
i N i C d 2  E .
As we see for rare gas at the high temperature the system is dominated by particles with high energy. Therefore they
are more confined and we have practical no overlaps. In such a case we can neglect terms with few particle in each
energy level. And the system obeys terms which allows many particles with the same energy and in limits we
converge to the classical statistic. On the other hand if energies are low only few particles are allowed in the same
energy level. And therefore the statistic is converging to Fermi-Dirac.
Boltzmann equation and transition to superconductivity. Boltzmann equation without external field
has the form

f ( p, x, t )  V .f ( p, x, t )  f coll , Where f ( p, x, t ) is the density of the particles with momentum p at the
t
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 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 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;

f ( p, x, t )  V .f ( p, x, t )  Cf ( p, x, t )  f imp  f def
t
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, x, t ) 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, x, t )  e Ct ( p, x, t ) where  ( p, x, t ) is the solution to the

f ( p, x, t )  V .f ( p, x, t )  0
t
It is easy to show that  ( p, x, t ) which can be constructed with the sinus and cosine base is a bounded solution if
 Ct
the initial value is sufficiently regular and bounded in the sense that e ( p, x, t )  0 as
the system tends to the state of superconductivity, if the temperature is low.
t   . And therefore
Integrity of the particles. In CQFT integrity of 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 external
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 existing in quantum theory. But also provide us with simple
methods of solving problems in many branches in physics.
References
1. D. F. Walls, G.J. Milburn “Quantum Optics”, Springer-Verlag 1994
2. Steven Weinberg “The Quantum Theory of Fields”Cambridge University press 1995.
3.G. Scharf, “Finite Quantum Electrodynamics”, Springer-Verlag 1995
4. E. Neother, Nachr. Akad. Wiss. Goettingen Math. Phys. Kl. p. 235 (1918)
5. Siegmund Brandt, Hans Dieter Dahmen “Quantum Optics”, Springer-Verlag 1995
6. H. Kamerlingh Onnes. (1911)
7. Neil W. Ashcroft, N. David Mermin “Solid State Physics”, Holt-Saunders International
Editions 1981
8. C. W. Kilmister, “Hamiltonian Dynamics.” Amer. Elsevier, New York, 1965.
9. Pierre Ramond, “Field Theory” FIP. Lecture Note Series, Addison-Wesley Publishing
Company, 1990.
10.Feynman, QED, PRINCETON UNIVERSITY PRESS (1985).
11.P.V.E. McClintock, “Low-Temperature Physics”, Blackie(1992)