Guidelines to Quantum Field Interactions in Vacuum

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.


Introduction
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 [1], we introduce some novelties in the derivation of final results. In particular, a discussion of the electron self-energy result in terms of a Zitterbewegung is presented.
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.

Particles and Fields
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 ( ) , x t Ψ depending on time and space coordinates t and x respectively. The wave functions are solutions of a differential equation known as the Schrödinger equation. In a heuristic way this equation can be derived by replacing the energy and momentum of the particle by operators, according to the relations  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 φ depending on the time-space vector 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 , x . We now define an action integral S over a region Ω bordered by a closed surface ( ) ∑ Ω , as follows: Varying this integral in the usual way according to the relation and using the identities , , The last term in the parenthesis can be seen as the four-divergence of a four-vector proportional to δφ . Therefore with Gauss's theorem it can be transformed into a surface integral over the border ( ) ∑ Ω . Since the Lagrange method postulates 0 δφ = at the surface, this term disappears. On the other hand, if the action integral S has to be an extremum, S δ must vanish for any value of δφ . This leads to the familiar Euler-Lagrange equations 0 , 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.
the Klein-Gordon equation The Hamiltonian.
In order to establish a link with classical mechanics, we first conceive the space coordinates i x as a countable set, each element occupying an infinite- Considering the classical expression of the Hamiltonian (2.20) 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  represents the Hamiltonian density 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 x space to p space by introducing the following transformations: The Hamiltonian density then takes the form }   3  3  2  3  3   d  d  1  e  2  2π 2π Since we want to quantize the system by replacing wave functions with operators in the Schrödinger picture, we disregard t in this expression.
Integrating over the space coordinates, we thus arrive at the following expression for the Hamiltonian in terms of functions in p space: Inserting into the commutator the transformation relations given By equation's (2.27a), (2.27c) we write 2π 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 p assumed to constitute an orthonormal set in a given inertial frame. Acting with the Hamiltonian of Equation (2.32) on one of these states, e.g.       Hence the transformed quantities are yielding the following result in terms of rotated quantities: Now the cumbersome factor 0 0 e ip ∆ has disappeared and the transformation ′ ′ ∆ → −∆ leaves the value of the second integral unchanged, since in this integral one can change the sign of the integration variable without affecting its value. The fact that for any point on a given curve the corresponding coordinate rotation can be made, and that this is true for any curve, proves the statement that the commutator vanishes at any point outside the light cone.
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. [1].
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.
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.

Symmetry Transformation Relations
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 , x x µ ν as J µν , and the corresponding rotation parameter as µν ω , then an infinitesimal transformation is generated by the operator yielding for the finite Lorentz transformation operator the expression we can generate a four dimensional representation of the proper Lorentz group by acting with this operator on the vector ( ) we consider the example 12 21 ω θ ω = = − , all other µν ω equal zero. Equation This matrix thus corresponds to a rotation by an infinitesimal angle θ in the xy plane as can be shown by multiplying the matrix by the vector ( ) As a second example we consider the Lorentz boost in the 1 x direction by setting 01 10 ω β ω = = − with all others equal zero. Then the relation Note that the factor 1 2 in Equation ( Applying a Lorentz transformation as expressed by the operator Λ of Equa- , one obtains the following change: 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 the scalar property of 2 φ is obvious. We therefore focus on the quantity ( ) where we have omitted on the r.h.s. the argument Note also that the horizontal shift of the lower indices on matrix elements allows us to distinguish between line and column indices. Since matrix elements are c-numbers, their product can be treated separately. It is sufficient to do this in the limit of infinitesimal rotations. The more abstract general treatment can be found in the literature e.g. in ref. [1].

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 λ is an infinitesimal quantity, we consider the expression g g In the first term the indices β and ν are eliminated yielding with β ρ = , ν β = =  g ρρ σ ρ λ ⋅ no summation whereas for the second term we find with ν σ Suppose now that ρ and σ belong both to ordinary space i.e. 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 , u u v v + − + − , with the + − signs indicating spin variables 1 2 + , 1 2 − in a given frame, the wave function in four space can be written in the form Considering components 1 u ψ + etc. as elements of a vector in spin space, we can also write where the functions 1 Ψ etc depend on both the space and the spin variable.
The column vector of Equation (3.17) is known as a spinor.
Its Lorentz transformation can be expressed as follows: ( ) where it is understood that the operator 1 2 Λ acts only on spin states.
We now define operator matrix elements S ρσ by introducing for 1 2 Λ the limiting expression with ρσ ω being the usual rotation and boost parameters.
We now recall that spin functions transform under rotations in ordinary space according to the Pauli spin matrices i σ with Then clearly, ordinary space rotations occur according to the relation  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 whereas the ij matrices remain unaffected.
showing that the functions L Ψ and R Ψ , called Weyl spinors, transform independently from each other.
Clearly, these relations can be generalized for arbitrary rotation and boost parameters described by vectors θ and β respectively. This leads to the transformation relations Hence the Weyl spinors as the corresponding Euler-Lagrange equation applied to Ψ , with the result Note that for µ µ γ ∂ and similar products Feynman has introduced the slash notation ∂ .
The γ matrices entering the Lagrange density are of vital importance, since in choosing them in an appropriate way, one meets the condition that  has to be a Lorentz scalar, necessary for the corresponding wave function to be valid.
As a consequence, there is clearly a connection between these matrices and the Lorentz transformation properties of the spinors. The corresponding relations are derived in many textbooks and will be given here only in their final form.
According to Dirac, the following equations hold: where the + index indicates an anticommutator. Note that later in this text the 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 L Ψ and R Ψ occurs because of the mass term in the Dirac equation.
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 Using the identity Introducing Noether currents by the defining relation

The Dirac Field
As an entrance door to the Dirac field let us consider free particle solutions of the Dirac Equation where for 0 γ the Weyl expression (3.35) has been used.
Introducing two-component spinors ξ , the solution is where the factor m has been chosen for future convenience. Let us now look for a more general solution with two components 0 p E = 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 η , called rapidity, replaces the previous β .
For finite values of η we therefore have The second expression on the r.h.s. is obtained by expanding the exponential and noticing that even powers of the matrix 0 1 1 0 whereas odd ones leave this matrix unchanged.
Now we apply the same boost to the amplitude ( ) Considering the matrix 3 Explicitating 3 σ and adding all matrices, a lengthy but straightforward calculation yields the following diagonal matrix where the relation has been used.
We now go back to Equation 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 x v p ⋅ Ψ = (4.14) We choose however to maintain 0 0 p > and hence   Replacing Ψ by the free-particle expressions of Equation's (4.1) and (4.14) we then have Introducing these expressions into Equation (4.25) yields the eigenvalue rela- Inverting the order of integration, we take advantage of the relation  Here the time dependence of the operators has been absorbed into the exponential factors. Moreover, the interchange † r r p p b b ↔ discussed above, has been taken into account.
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.

Propagators
The retarded Green's function.
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 expres- whereas for the upper circuit, corresponding to 0 0 x y < , the integral is zero.
This can also be written as where T is the time ordering operator which ensures that the earlier time always stands on the right, with the additional condition of a minus sign if the operators are interchanged.
In the Feynman case the integration paths can be slightly modified with respect to those of Figure 2 2π Performing the integration over 0 q along the paths indicated in Figure 2

Interacting Fields: The Radiative Electron Mass Shift
Introduction.
Consider an electron in the form of a point charge-e, then the surrounding static electric field possesses the energy  In this way during the process the total momentum of the system is conserved at every step. In addition the expressions describing the electron-photon interaction have to be inserted at the vertices.
In these expressions the Feynman slash notation abbreviates the sums p µ µ γ and k µ µ γ , whereas g µν is the metric tensor represented by a 4 dimensional diagonal matrix with 00 1 g = , 11 . 1 etc g = − .
Assembling these relations, known as the Feynman rules, we see that the above diagram corresponds to the product where now all matrices are replaced by scalars.
Writing out explicitly the product of (6.23) we obtain from the defining rela- 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 ( ) x δ corresponding to the non relativistic case. We want however to emphasize that the electron is still regarded as a point particle, but one that giggles around some central position producing an apparent spread of its mass.

The Electron-Electron Scattering (M ∅ LLER) Amplitude and Its Yukawa Analog
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 p and k respectively and outgoing momenta p′ and k′ , momentum conservation demands that matrix elements of iT satisfy the relation ( ) ( ) 4 4 , , 2π p k iT p k p k p k i δ where i is the scattering amplitude which is of interest here. In the fully quantized theory interaction takes place by means of the exchange of a virtual particle of momentum q.
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 16) The non-relativistic limit.
In the non relativistic limit where it is assumed that the kinetic energy of the For the sake of completeness we indicate the link between the amplitude  and the differential cross section. In the center of mass frame the following relation holds: Substituting for  the expression (7.22) we thus obtain 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 m , i.e. nucleons, and for the binding the photon is replaced by a massive particle of mass m φ much smaller than m called meson.
The calculation can be deduced from the previous one by replacing the photon propagator by the meson propagator Although the Yukawa model has been replaced since by more evolved concepts, it still provides insight into the nature of nuclear forces.

Vacuum Polarization
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 where terms linear in l have been omitted.
In [1]  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 Figure 4, which therefore has to be replaced by the configuration of Figure 5. One then expects that the global effect corresponds to the scalar quantity   which is indeed the value found in the literature.

Atomic energy level shift
Consider now the Coulomb potential as given in q space by Equation (7.25).
Its modification due to vacuum polarization produces a relative change equal to showing that only s levels will be affected. In the case of hydrogen the effect represents a small part of the Lamb shift. Larger effects can be predicted in the case of muonic atoms, i.e. atoms where the electrons are replaced by µ mesons [9].
For numerical values of the expected or measured shifts we are referring to the abundant literature on this subject.

Conclusion
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. Interac-tions 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.