In this treatise we stress the analogy between strongly interacting many-body systems and elementary particle physics in the context of Quantum Field Theory (QFT). The common denominator between these two branches of theoretical physics is the Green’s function or propagator, which is the key for solving specific problems. Here we are concentrating on the vacuum, its excitations and its interaction with electron and photon fields.
It is the aim of this treatise to pay tribute to Feynman’s propagator method and its visualization in Feynman diagrams. This method has applications as wide as e.g. many electron theories, condensed matter physics and quantum field theory.
It consists on one hand of showing for intricate mathematical expressions of the underlying physics, and on the other hand, of applying pre-established rules to these graphs, to set up these expressions.
Here we are not giving a lecture on these procedures; we are merely applying them to vacuum excitations interacting with electron and photon fields.
Starting from routinely used techniques as e.g. developed in the book by M. E. Peskin and D. V. Schroeder [
In a first introductory part, we recall the basic facts of the second quantization of the Klein-Gordon and the Dirac field and discuss the resulting consequences.
Then we define propagators for the Dirac and photon fields and use them to treat interactions of these fields with the vacuum. More specifically we study the electron and photon self-energies.
We do not concern ourselves in general with collisions between elementary particles, although this is one of the main subjects met in Quantum Field Theory. As an exception we consider however electron-electron scattering because of its connection with vacuum polarization. The resulting physical facts are discussed extensively.
It is the aim of this section to recall how, in relativistic quantum physics, negative energy states are avoided by adopting the field viewpoint. For this purpose we chose as the simplest possible case that of an uncharged particle obeying the Klein-Gordon equation. The essential arguments developed here then apply equally to the case of more general systems.
Negative energy states, causality.
In quantum mechanics we associate a particle with a wave function
For a particle we then have in the non relativistic case
In the relativistic case we start from the relation
This relativistic version of the Schrödinger equation is called the Klein-Gordon equation. It is important to note that in contrast to the non relativistic Equation (2.1) the Klein-Gordon equation contains the second time derivative meaning that it allows for negative energy solutions. Using from now on natural units
Setting
Equation (2.3) reduces to
where we have used
For plane wave solutions with
Hence there are negative energy solutions. The question arises whether these solutions cannot be discarded as non physical. But in that case we would not have a complete set of basic functions since these solutions are part of it. In actual calculations this could yield erroneous results. Furthermore, in a less obvious way, omitting these solutions leads to a violation of the principle of causality as we shall demonstrate now.
Consider the amplitude
Inserting the wave functions
we have
Using polar coordinates as follows:
we arrive after integration over
For simplicity we set
up to a rational function of
Bessel function reduces essentially to the exponential
for
Given this factor in the expression of
There are however other shortcomings contained in the relativistic particle theory. One could argue that any positive energy state must be unstable since after some time the particle would fall into a lower energy state, in the same way as an atomic electron in an excited state falls into the ground state after some short lifetime. In the case of fermions this can be prevented by assuming, following Dirac, that all negative energy states are occupied already. This situation is due to the fact that, according to the Pauli principle, each state can only receive one electron. The completely filled negative states constitute the Dirac sea. Moreover, this picture has led Dirac to the prediction of the positron, i.e. a positively charged electron, appearing as a hole in the Dirac sea when by some process an electron is removed from it.
It is however possible to give a less artificial description of relativistic quantum particles by adopting the field viewpoint which will be presented now.
Lagrangian field method
We consider a field function
relation with
indices belong to the Minkowski four-space, Latin ones to ordinary space, with
In analogy with classical mechanics, we introduce a Lagrange function, having here the character of a density, given by the expression
Note also the complementary relation
action integral
Varying this integral in the usual way according to the relation
and using the identities
we arrive at
The last term in the parenthesis can be seen as the four-divergence of a four-vector proportional to
or more explicitly
These equations apply to classical fields, e.g. one component of the electromagnetic vector potential, as well as to wave functions in particle quantum mechanics.
As an example let us therefore consider the Klein-Gordon wave function.
Setting
we write
yielding with
The Hamiltonian.
In order to establish a link with classical mechanics, we first conceive the space coordinates
Considering the classical expression of the Hamiltonian
with the canonical variable
we have the correspondence
defining the canonical variable
With these definitions we obtain for the classical relation (2.18) the following equivalent expression:
Switching now to the limit of continuous space coordinates, this result takes the form
where
with
Let us consider as an example the Klein-Gordon case.
According to Equation (2.16) the Lagrange density can be written as
We then have
Second quantization.
Simply speaking, a given wave function is quantized if it is replaced by an operator. This is familiar in quantum electrodynamics where e.g. one component of the vector potential is replaced by photon creation and annihilation operators. A similar procedure can be applied to quantum mechanical wave functions and in this latter case one then talks of second quantization, since the wave functions are already obtained by a first quantization procedure. Note however that the term second quantization is not universally accepted.
Here we consider again as an example the Klein-Gordon case, which constitutes the simplest one, as it concerns spinless particles like K or π mesons.
Let us first switch from
The Hamiltonian density then takes the form
Since we want to quantize the system by replacing wave functions with operators in the Schrödinger picture, we disregard
Integrating over the space coordinates, we thus arrive at the following expression for the Hamiltonian in terms of functions in
with
To obtain Equation (2.29) we have made use of the relation
The parenthesis inside the integral of Equation (2.29) reminds one of the Hamiltonian
of a harmonic oscillator.
In the latter case quantization is achieved by introducing creation and destruction operators
with the commutator
We therefore try in Equation (2.29) the substitutions
The parenthesis inside the integral in Equation (2.29) is then found to be given by the expression
Since complete summation over
We thus obtain for the Hamiltonian the following result
According to general rules of quantum physics, the commutation relation for canonical variables takes the following form in the present case:
Inserting into the commutator the transformation relations given By equation’s (2.27a), (2.27c) we write
Substituting for
Adopting the trial rule
Equation (2.35) reduces to
Substituting this result into Equation (2.34) we recover the commutation relation of Equation (2.33). This confirms the validity of the trial rule of Equation (2.36).
In the field equations developed above the number of particles concerned is not specified. Let us now be more specific by introducing single particle states
The second term on the r.h.s. of this equation contains the infinite quantity
In order to establish the time dependence of the operators
Starting from the expressions (2.31a), (2.31b) we evaluate the corresponding Heisenberg operators of
Acting on an eigenstate
using
Similarly we have
Hence the requested operator equations are
With Equation (2.31b) the quantized form of Equation (2.27a) becomes
where Equation’s (2.39a), (2.39b) have been used.
Introducing the Lorentz invariant scalar product
Causality again.
As mentioned earlier, two points
Starting from Equation (2.41) the commutator is given by the expression
where the operator commutation rule of Equation (2.36) has been used. In order to obtain zero for this quantity, the inversion transformation
Now we define a space like surface [
Without loss of generality we can restrict ourselves to the plane
Now take a particular point
One then has the relations
Hence the transformed quantities are
yielding the following result in terms of rotated quantities:
Now the cumbersome factor
Inside the light cone, i.e. for time like separations, the commutator does not vanish so that in this region points can be causally connected. It is however interesting to note that the corresponding commutator is invariant with respect to proper Lorentz transformations as shown e.g. in ref. [
Note finally, that in many calculations the infinite energy of the vacuum state is eliminated by performing normal ordering of operators. It consists in reshuffling operator products in such a way that destruction operators always stand on the right of creation operators.
Generalizations [
Particles obeying the Klein-Gordon equation do not bear any electric charges. In order to treat charged particles, complex wave functions have to be introduced into the theory. Even more profound modifications are necessary in the case of electrons according to the Dirac theory. Here, due to the presence of spin, wave functions are represented by spinors consisting of four functions as components of a vector. An even more striking difference occurs if second quantization is performed. In this case, the fermion character of the particle is taken into account in postulating anti-commutation rules for the field operators instead of the commutation rules pertaining to bosons.
However, the general idea of avoiding negative energy states by means of second quantization, already applied to the Klein-Gordon case, remains essentially the same in this and other situations.
An essential feature of relativistic particles and fields is their behaviour with respect to transformations of the Lorentz group.
Transformation operators
We recall that the elements of this group are three rotations in the xy, xz, and yz planes around the z, y and x axis respectively, completed by three pseudo-rotations belonging to the xt, yt and zt planes respectively. These transformations can be viewed as an infinite succession of infinitesimally small rotations which generate a representation of the group. Designating the rotation operator with respect to the plane
yielding for the finite Lorentz transformation operator the expression
Recalling that the familiar expression for rotations in ordinary space can be generalized to Minkowski space as
we can generate a four dimensional representation of the proper Lorentz group by acting with this operator on the vector
, (3.4)
we consider the example
This matrix thus corresponds to a rotation by an infinitesimal angle
As a second example we consider the Lorentz boost in the
with
Note that the factor
of Equation’s (3.5) and (3.7) by the column vector
relations for the corresponding infinitesimal rotations and Lorentz boosts.
Applying a Lorentz transformation as expressed by the operator
The criterion for the corresponding wave equations to be valid is their Lorentz invariance. This property can be established by proving that the Lagrange density, from which a given wave equation is derived, is a Lorentz scalar. We shall now demonstrate this point in the particular case of the Klein-Gordon equation.
We cast the Lagrange density of Equation (2.16) in the form
with only one type of differential operator. With the transformation of Equation (3.8), i.e.
the scalar property of
where we have omitted on the r.h.s. the argument
According to Equation’s (3.1) and (3.2) we write
With the defining relation
Treating only the change introduced by the transformation and given the fact that
In the first term the indices
whereas for the second term we find with
Hence the final result
Suppose now that
The proof given here for infinitesimal variations is generally valid, since finite transformations involve an infinite succession of infinitesimal ones. As already mentioned, more formal proofs are found in the literature, but we thought it instructive to approach the problem by explicit calculations as well.
Spinors.
Having treated as an example the case of a structure less particle obeying the Klein-Gordon equation, we are now moving to the case of the electron, where in addition to space coordinates spin variables have to be considered, together with the existence of an electric charge.
Introducing spin functions
Considering components
where the functions
Its Lorentz transformation can be expressed as follows:
where it is understood that the operator
We now define operator matrix elements
with
We now recall that spin functions transform under rotations in ordinary space according to the Pauli spin matrices
Then clearly, ordinary space rotations occur according to the relation
i j k in normal order.
Remark: normal order means that i j k are all different and that starting with 1 2 3 an odd number of permutations introduces a minus sign. One may ensure this property automatically by multiplying with a quantity known as the
The question now arises, what happens in the case of Lorentz boosts? Without entering into details, we only state the answer given by Dirac’s theory according to the relation
Hence the matrices of Equation’s (3.21) and (3.22) constitute a four-dimen- sional representation of the Lorentz group known as the Dirac-Pauli representation.
The Weyl representation.
The Dirac-Pauli representation is reducible since its matrices can be brought into diagonal form by a unitary transformation involving the matrices
With these matrices we have
and hence
whereas the
Designating as left and right handed spinors
Taking as an example the values
showing that the functions
Clearly, these relations can be generalized for arbitrary rotation and boost parameters described by vectors
Hence the Weyl spinors
In order to explain the designations of
spinors.
As an example the spinors introduced in Section 4 are right handed for those of Equation’s (4.13a), (4.15a) and left handed for those of Equation’s (4.13b), (4.15b). This can be shown by applying the helicity operator with
Connection with wave equations.
The wave equation for spinors
as the corresponding Euler-Lagrange equation applied to
Note that for
The
where the + index indicates an anticommutator. Note that later in this text the anticommutator will be designated by the symbol
Given the fact that the matrices
Making the guess that
one obtains the result
By setting
one then obtains the following relations:
The Dirac equation, given in its general form by Equation (3.29), then takes in the case of the Weyl representation the form of the following two coupled equations:
written in matrix form as
As can be seen from these equations, the mixing of the two Lorentz group representations
Noether currents.
Let us now consider some continuous symmetry transformations on the wave functions, which leave the Lagrangian density invariant. In the infinitesimal limit we then write
The corresponding change in the Lagrange density
With the obvious relation
we then have
Using the identity
the second term on the r.h.s. of Equation (3.41) can be rewritten with the result
Now the second term of this equation, set equal to zero, represents the Euler-Lagrange equation as given by Equation (3.14). For
Introducing Noether currents by the defining relation
Equation (2.43) involves the four-divergence of this quantity for which we thus have
Integrating this expression over the entire ordinary space, and applying Gauss’ theorem to the corresponding three-divergence, with vanishing contribution at the infinite surface, we are left with the expression
Hence the space integral
In order to interpret this quantity, let us consider the Dirac equation. The corresponding Lagrange density function is given by Equation (3.28). This equation is invariant under the phase transformation
For Noether’s current we then have, according to Equation (3.44)
and
where we have used the fact that in any representation
As an entrance door to the Dirac field let us consider free particle solutions of the Dirac Equation (3.29). These solutions can be viewed as superpositions of plane waves of the form
Plugging this expression into Equation (3.29), yields the equation
This equation is most easily solved in the rest frame, where only the component
where for
Introducing two-component spinors
where the factor
Let us now look for a more general solution with two components
This solution can be obtained by performing a Lorentz boost on the previous one, which in infinitesimal form can be written as
This relation can be deduced by analogy from the matrix of Equation (3.7) noticing that all spatial directions are equivalent whereas the infinitesimal parameter
For finite values of
The second expression on the r.h.s. is obtained by expanding the exponential
and noticing that even powers of the matrix
whereas odd ones leave this matrix unchanged.
Now we apply the same boost to the amplitude
From the infinitesimal operator as given by Equation (3.24) with I = 3, we deduce the relevant Lorentz transformation operator
Considering the matrix
power of the matrix in the exponent of Equation (4.9) yields the unit matrix, whereas an odd one yields this same matrix. The series expansion of the exponential operator of Equation (4.9) therefore leads to the following matrix expression:
Explicitating
where the relation
has been used.
We now go back to Equation (4.8) and calculate the amplitude
special spinors
positive and negative
So far we have put the minus sign on the exponent of the defining relation given by Equation (4.1). Consider now the case of a plus sign with
We choose however to maintain
We are not repeating a calculation similar to the previous one, but indicate only the relations replacing Equation’s (4.13a), (4.13b). For these special situations one finds
Defining as usual
With
A similar calculation for the case of Equation (4.15b) yields the result
For the case of an arbitrary spin orientation axis we introduce the notations
Furthermore we have the relations
, (4.20)
The Hamiltonian.
Starting from the expression (3.28) of the Lagrangian density
and from the expression of the conjugate variable
density is given, according to Equation (2.23) by the expression
More explicitly we then have with
In the expression of
Involving the single particle Hamiltonian
The amplitudes
remembering that
This equation can be expressed in the form
Replacing
Introducing these expressions into Equation (4.25) yields the eigenvalue relations stated above
with
Second quantization.
In replacing the wave function
or equivalently
Defining an empty state
Introducing the total Hamiltonian
After substituting the expression (4.29) and its adjoint we write
Inverting the order of integration, we take advantage of the relation
and notice that, according to Equation (4.20), the cross terms in the product of the integrand in Equation (4.31) disappear. We are thus left with the expression
where, given the integration over all values of
Eliminating the amplitudes by means of the relations (4.18), (4.19), we thus arrive at the final expression
At this stage it has to be reminded that in the present case of fermions the operators obey anti-commutation relations, which in contrast to the boson relations (2.37), are of the form
This relation allows us to deal with the embarrassing negative energy term in the integrand of Equation (4.33).
Writing by means of the rule stated by Equation (4.34)
we have cast the negative energy into an infinite constant term which can be ignored if the origin of the energy scale is shifted adequately.
A next step consists in interchanging the order of
Normal ordering
A procedure of eliminating negative energy terms in the Hamiltonian consists in what is called normal ordering. It means that all operator products are reshuffled in such a way that annihilation operators stand always on the right of creation operators. These operations are symbolically expressed by the letter
Applying this convention to the expression (4.22), supposed second quantized, we thus write
where
Here the time dependence of the operators has been absorbed into the exponential factors. Moreover, the interchange
A calculation similar to that developed above, with only the cross terms contributing, then leads to the expression
This is exactly the result obtained previously if in Equation (4.35) the infinite negative energy term is ignored and if the operator and state changes discussed there, are accomplished. Thus clearly normal ordering merely integrates these facts.
The retarded Green’s function.
Let us first consider propagation amplitudes given by the expressions
These expressions are obtained by using the fact that in the product of the wave functions of Equation’s (4.37a), (4.37b) only cross terms contribute. This is because in the other terms annihilation operators are on the right and therefore eliminate these terms in the mean values of Equation’s (5.1a), (5.1b). Furthermore the operator relation (4.34) has been accounted for.
We now evaluate the spin sums appearing in Equation’s (5.1a), (5.1b). Using Equation’s (4.13a), (4.13b) and (4.15a), (4.15b) we obtain the following tensor products:
where the relation
A similar calculation yields
For the spin sum we therefore arrive at the result
It is now an easy matter to show that this matrix is identical with the expression
with after a similar calculation
Making these replacements in the expressions (5.1a), (5.1b) and adding them afterwards, we obtain an anticommutator of the form
with
We now want to link the above commutator to an integral in four space. For this purpose we introduce a quantity defined by the relation
where
for
for the lower clockwise circuit, corresponding to
whereas for the upper circuit, corresponding to
Inserting the value given by Equation (5.7) into the complete integral given by Equation (5.6) we can, without loss of generality, replace in the second term
and for
Comparing with Equation (5.4) we thus find
where
Going back to Equation (5.6) we notice that the denominator
Written in the form
this quantity can be regarded as the Fourier transform of
or in Feynman slash notation
This expression is known as the Dirac propagator. Its Fourier transform represented by Equation (5.10) is a Green’s function of the Dirac operator defined in Equation (3.29). To see this, we first notice that for plane wave states this operator can be written as
thus proving the Green’s function relation stated above.
Note however that the integral in Equation (5.10) can be evaluated along different paths. The way chosen so far yields the particular expression (5.9), called the retarded Green’s function. This is because it is only non zero during the time period
The Feynman propagator.
A different path for evaluating the integral of Equation (5.6) is that shown on
Inserting these expressions into Equation (5.6) we obtain the Feynman Green’s function
where again in the second line
Comparing these expressions with Equation (5.4) we see that we have
This can also be written as
where
In the Feynman case the integration paths can be slightly modified with respect to those of
With the denominator equal to
away from the real axis to
of the integration paths.
Interpreting Equation (5.18) as a Fourier integral we thus obtain for the Feynman propagator the expression
or in slash notation
These expressions are basic elements in Many-Body type calculations.
The photon propagator.
In analogy with Equation’s (5.16) and (5.17) representing the Feynman propagator in the Dirac case, we define a photon propagator by the relations
Corresponding to the time-ordered product
Here
The quantities
Postulating the rule
This expression reduces to
with
The value of the quantity
As in the Dirac case we now link the expression (5.25) to an integral in 4 dimensional space of the following form:
Performing the integration over
Comparing this with Equation (5.20) we see that the two integrals correspond to the expressions defining the propagator
Setting
as the final result for the photon propagator in the Lorentz-Feynman gauge.
Introduction.
Consider an electron in the form of a point charge-e, then the surrounding static electric field possesses the energy
with
out this treatise we use natural units setting
In order to make the integral in Equation (6.1) finite, a lower cut-off radius
In this way the energy tends linearly towards infinity with the cut-off parameter
Attempts have been made to improve things by applying the formalism of quantum field theory to this problem. In this treatise we present a slightly renewed version of these calculations. As a result the linear divergence of the semi-classical theory is brought to the form of a logarithmic one however with no quantitative solution at the end.
The propagators.
Preliminary remark: as is customary in quantum field theory we designate vectors and indices in 4 dimensional Minkowski space by ordinary letters and l.c. greek letters (e.g.
We now consider an electron moving freely through vacuum and define a correlation function by the expression
where
In expression (6.3) the Dyson operator
The presence of ground states
The easiest way for evaluating the correlation function (6.3) consists in applying Feynman rules according to the Feynman diagram of the figure (
The elements of this diagram correspond to Feynman propagators in momentum space given by the expressions
for the electron of mass m in momentum state p and k respectively and the propagator expression
for the photon.
In this way during the process the total momentum of the system is conserved at every step. In addition the expressions
, (6.6)
describing the electron-photon interaction have to be inserted at the vertices.
In these expressions the Feynman slash notation abbreviates the sums
Assembling these relations, known as the Feynman rules, we see that the above diagram corresponds to the product
where the relation
where the central part is given by the expression
after adding an integration over all possible intermediate 4 momenta.
The index on
The integration procedure.
Before starting the integration in the expression for
Comparing with Equation (6.9) we thus write the
Following a common procedure we now change variables according to the relation
Then the parenthesis in the denominator of the integrand takes the form
An essential simplification arises if we restrict ourselves to the zero’th order contribution in
The integral in (6.10) then reduces to
Note that the same letter
Separating the
where now all matrices are replaced by scalars.
The evaluation of the second integral is presented in Appendix leading to the result
Setting
Introducing the dimensionless variable
with the limiting expression
where we have assumed that the cut-off value
Plugging this result into Equation (6.9) we thus arrive at the final expression
Renormalization.
Let us suppose that the change in the correlation function represented by the resulting expression (6.19) can be reproduced by renormalizing the mass in the free electron propagator, i.e. by adding a correction
Equating the correction term with the expression (6.8) with the expression (6.19) for
Approximating on the r.h.s.
This is the result derived in the literature by various methods, showing that the fully quantized theory reduces the linear convergence of the classical expression (6.2) to a logarithmic one.
Discussion.
There seems to be no indication how to estimate the cut-off parameter
a number that seems realistic. Naturally this estimation has to be taken merely as an example among others that one could imagine.
However, despite the fact that the true numerical value of the electromagnetic electron mass shift is as yet unknown, its correct qualitative evaluation, as reviewed in this section undoubtedly constitutes an important fact.
Zitterbewegung
The fact that quantum field calculations lead to a logarithmic divergence of the electron self-energy instead of the linear classical result of Equation (6.2), can be understood if one takes into account the spread of the electron position due to quantum fluctuations [
In order to determine the position
Interpreting the central part
as a density operator we obtain the desired average by taking a trace represented fomally by the expression
Writing out explicitly the product of (6.23) we obtain from the defining relation (5.14) the result
under the condition
Furthermore, the variable
The taking of the trace in Equation (6.24a) amounts to integrating over the variable y and afterwards replacing the matrix
The integral of Equation (6.27) is elementary, yielding with
With the last integral on the r.h.s being equal to
probability the final result
As a test we integrate over the entire space and find
thus proving the validity of our probability calculation.
Clearly a distribution as represented by Equation (6.29) will lead to a softer divergence than one of the type
Consider scattering involving two particles and introduce a scattering matrix in the form
where the second term describes the scattering process.
Assuming that the particles have incident momenta
where
We specialize now to the case of two colliding electrons schematically represented by the Feynman diagram below.
We write the Hamiltonian of the system in the form
where
with
The relevant contribution here is the second order term in the perturbation expansion of the
Given the interaction Hamiltonian of Equation (7.4) this term contains the time-ordered product
with T the familiar time ordering operator. Note that a factor 1/2 from the exponential expansion is left out since it is compensated for by adding identical expressions with
Substituting into the parenthesis the expressions derived in section (3) for
At this stage we suppress for simplicity spin labels on the operators and functions.
Putting in the expression (7.7) the operators in normal order we make the replacement
We now take matrix elements between states
Together with the preceding sequence of Equation (7.8) this generates the new operator sequence
We now make use of operator commutation relations which yield the equations
similarly for
Now after integrating in Equation (7.7) over the variables
Going back to Equation (7.6) and recalling that the contraction of vector potential operators is equivalent with the propagator expression
we obtain the matrix element in the form
Identifying the
Comparing this expression with the defining relation (7.2) we find for the electron-electron scattering amplitude the formal expression
The non-relativistic limit.
In the non relativistic limit where it is assumed that the kinetic energy of the electrons is small as compared to
, (7.17)
with
Then for
, (7.18)
with
The products in Equation (7.16) are
Furthermore we have in this approximation with
Thus the amplitude of Equation (7.16) reduces to
Now clearly, labeling the spins by
Consider now the electrostatic potential
In the case of a Coulomb potential
An elementary integration yields the result
Comparing this result with Equation (7.22) one sees that the amplitude factor
For the sake of completeness we indicate the link between the amplitude
Substituting for
Note that this expression is equal to the celebrated Rutherford formula which applies to scattering of a particle in a static Coulomb field.
The Yukawa potential.
An approach similar to that leading to the Coulomb potential, treated in terms of the exchange of a photon between two electrons, has been proposed by Yukawa in 1935 for the interpretation of nuclear forces. Here the interaction takes place between heavy particles of mass
The calculation can be deduced from the previous one by replacing the photon
propagator by the meson propagator
of the meson and the electro-magnetic interaction
In the non-relativistic limit one finds
Connecting in this limit the scattering amplitude to the potential
with in the
Setting
This attractive potential is short ranged as compared with the Coulomb potential. The presence of the exponential factor yields for this range the value
Although the Yukawa model has been replaced since by more evolved concepts, it still provides insight into the nature of nuclear forces.
The photon self energy.
Consider a photon propagating freely in vacuum. If its interaction with the vacuum field is taken into account, a situation represented by the Feynman diagram below will be present. During the propagation there will be emission/absorption of a virtual electron/positron pair at one vertex and afterwards the inverse process will occur at the other vertex.
The difference with respect to the case without interaction involves a tensor which in second order will be written as
Applying, as in the electron case, the Feynman trick and setting afterwards
one arrives at the expression
where terms linear in
In [
with
This integral is ultraviolet diverging. It can be simplified by using the tensorial relation
involving the scalar quantity
Assuming now
For the integral on the r.h.s. we have, according to [
The remaining integral is logarithmically ultraviolet diverging. Let us calculate it however formally as follows:
Pauli-Villars regularization.
The Pauli-Villars regularization consists in making the integral convergent by subtracting the same expression but with
The integral of Equation (8.9) thus becomes
For the quantity of interest we therefore find
Considering
with
Charge renormalization.
Going back to the electron-electron scattering problem clearly the photon self-energy effect just discussed, will manifest itself as a modification of the photon propagator represented by the wavy line in
The amended Coulomb potential
Having treated the diverging expression in (8.7) by means of a regularization procedure, we are now going to extract from this expression a term which is independent of any cut-off parameter. For this purpose we make the following first order expansion:
where we have set
assuming
where the equivalence
Expliciting now
With the values of the integrals equal respectively to
find
which is indeed the value found in the literature.
Atomic energy level shift
Consider now the Coulomb potential as given in
Taking the inverse Fourier transform yields for the amended potential in
Applying this potential to electrons inside an atom will lead to a shift of energy levels obtained by multiplying the correction term with the electron density function and space integration. The effect then becomes proportional to
For numerical values of the expected or measured shifts we are referring to the abundant literature on this subject.
In this treatise we are interested in phenomena involving the presence of what is sometimes called the physical vacuum. To deal with these effects, one adopts the field viewpoint, which consists of replacing for elementary particles, e.g. electrons, wave functions by operators acting on physical vacuum states. Interactions between fields defined in this way are then treated according to Feynman’s propagator method. The main difficulty affecting this method is the appearance of divergencies which are dealt with by means of two specific procedures known as regularization and renormalization. The first one consists of making expressions finite by applying e.g. cut-off or Pauli-Villars regularization. The second one is a redefinition of physical quantities, e.g. electric charge or mass, in accordance with the finite results previously obtained. In this treatise, we consider mainly results for the electron self-energy and the vacuum polarization case. Some of our derivations of these results are original and special attention is given to their interpretation in terms of the underlying physical facts.
Particular thanks go to Prof. Gillian Peach and to Prof. Cynthia Kolb Whitney for reading and improving the manuscript.
Schuller, F., Neumann-Spallart, M. and Savalle, R. (2017) Guidelines to Quantum Field Interactions in Vacuum. Journal of Modern Physics, 8, 382-424. https://doi.org/10.4236/jmp.2017.83026
Evaluation of the integral
Setting
With the change of variables
The integral in (A2) takes the form
where we have deliberately not specified the integration limits.
Introducing the identity
we ignore the principal value which in a more detailed treatment can be proven to yield zero. With the delta function inserted the expression (A4) then reduces to
Performing the derivation as indicated in Equation (A2) and replacing the intermediate parameter A by its value leads to the desired result