Feynman, R. P. (1949) The theory of positrons. Physical Review, 76 (6). pp. 749-759.
https://authors.library.caltech.edu/3520/
Feynman, R. P. (1949) Space-time approach to quantum electrodynamics. Physical Review, 76 (6). pp. 769-789.
https://authors.library.caltech.edu/3523/
Feynman's second 1949 paper considers the path of two electrons in a scattering experiment.
The starting point is no interaction. The two electrons trace their path as free particles according to the Dirac equation. The free particle case is easy to handle: the wave function is the direct product of two free particle wave functions.
Then Feynman adds the repulsive Coulomb force that only is switched on for a short time dt. The Hamiltonian is modified for the interval dt, which creates a correction term to the original wave function. The term is small. That is why this is called the perturbative method.
By integrating over all dt we get a correction term for the interaction for the whole time interval. The correction term is now bigger, but we still use the, perhaps misleading, name perturbation.
An electric field invokes vacuum polarization. The electric field is slightly distorted because it creates a disturbance in the Dirac field. In the Feynman formulas, the disturbance is modeled by a term which can be interpreted as the creation and annihilation of a virtual electron-positron pair.
Unfortunately, the term diverges because it allows an arbitrary value for the momentum p of the virtual pair. If we limit p to some sensible interval, we get finite values and can do extremely accurate calculations of the real physical process. This is called renormalization.
The goal of this blog post is to understand intuitively why the divergence happens and what it means.
If we modify a wave function, its integrated probability over all possible outcomes should stay as 1.
Why the Feynman loop term does not respect this rule? If the term becomes infinite, then it breaks the rule maximally.
It looks like there is some error in the Feynman formula in the first place. Maybe the probability amplitude associated with large values of p is exaggerated?
If we take the Feynman diagram literally, the probabilities associated with each path of real and virtual particles should add up to 1.
Let us consider a simpler case: the double slit experiment.
We trace all possible paths for the photon. One half of the paths go through slit A and the rest through slit B. The interference pattern on the screen adds up the complex number probability amplitudes. A constructive interference and a destructive interference alternate. The alternation makes the integral of probabilities over the whole screen as 1.
If we would use some approximation method in the calculation of the interference pattern, the probability integral might become infinite if we either inflate the amplitudes in some path or exaggerate greatly the constructive interference. Which is the case in the Feynman loop integrals?
Self-energy of an electron
Feynman integrals involve an "infrared" divergence in the "self-energy" graph, where an electron sends a virtual photon to itself.
~~~~~ virtual photon
/ \
e- --------->-------------
For a free electron, hypothetical virtual self-energy photons are not relevant. We can describe a free electron with the Dirac equation.
But when an electron is accelerated, then self-energy has an impact. What is this self-energy? Our discussion of the Larmor formula offers a clue:
http://meta-phys-thoughts.blogspot.fi/2018/04/the-error-in-larmor-formula-for.html
Radiation is produced by the interaction of charges. The charges together cause a disturbance to the electromagnetic field. Far away from charges the field is under an essentially linear wave equation and its disturbance has a Fourier decomposition. Far away, we can talk about electromagnetic radiation and photons.
But close to the charges, the system is nonlinear. The interaction between the field and the charges is complex. Momentum and energy can be transferred to the field and back to the charges in a complex way. This explains why an electron can move momentum and energy to the field and receive it back later.
Thus, the self-energy is nothing mystical but a regular phenomenon of classical electrodynamics. In classical electrodynamics the amount of momentum transferred from the electron to the field is not arbitrarily large, as the Feynman formulas postulate, but restricted by the geometry of the situation.
Open problem 1. Is there a reason why Feynman formulas should allow an arbitrary momentum for the self-energy virtual photon?
The following analogue might clarify a nonlinear scattering process. If we have two people strirring water with paddles that are close to each other, then the interaction between paddles is mediated mostly by water currents. But far away, the movement of water can be approximated by waves. The interaction to a far away paddle would be mediated by waves. "Virtual photons" are the water currents while real photons are waves.
The name "virtual photon" is misleading. A better name would be a direct force impulse. A force impulse can be between charges or between a charge and the electromagnetic field.
We have a hypothesis that electromagnetic waves are just polarization of virtual electron-positron pairs. In this framework, a direct force impulse is always between electrons or positrons, virtual or real. The electromagnetic field is just an abstraction.
Let us then try to decipher where and why arbitrarily large momentums enter the Feynman calculations. Feynman uses Green's functions to approximate solutions to an inhomogeneous linear differential equation. Inhomogeneous means that we have a linear system which is disturbed by a "source" function. An example is a string which is plucked.
A Green's function describes an infinitely sharp "point impulse" to a wave equation and the diffusion of the impulse afterwards. The point impulse is a Dirac delta function which starts to develop in time and diffuse according to the wave equation.
A Dirac delta function can be built by adding an infinite number of plane waves where all waves have a constructive interference at time t = 0 at position x = 0. The waves have arbitrary momentum p. This is obviously the source of arbitrary momentums in the Feynman formulas.
The question is whether allowing arbitrary momentums makes sense in the nonlinear system we are studying.
The poles in the energy-momentum representation
In The theory of positrons, section 6, Feynman says that the Fourier transform of K_+(2, 1) is of the form given in his equation (31). The integral in (31) contains (p - m)^-1, which introduces two poles in the integral, at E and -E in the notation of Feynman.
The poles are the first symptom of diverging integrals in Feynman rules. Feynman handles the poles by adding an infinitesimal imaginary mass to m, which moves the poles out of the real axis.
Let us try to decipher what is the origin of these poles.
The notation p - m is not clear. The p is apparently a 4 x 4 matrix built by summing gamma matrices for each of the 4 components of the momentum of the electron. The fourth component is apparently the mass-energy "moving" forward in time.
But what is m? Do we multiply the fourth gamma matrix with the rest mass of the electron to obtain m?
What is the physical intuition why the Fourier decomposition of K_+(2, 1) should contain an infinite component when p - m happens to be a zero matrix?
Why handling the poles by adding an infinitesimal imaginary part to m produces sensible results?
The Dirac equation with a potential is
(i∇ - m) ψ = A ψ.
Let the potential A = 0. We can "disturb" the equation with a "source" pulse
(i∇ - m) ψ = δ(x - x_1)
at a spacetime point x_1.
Feynman uses a Green's function to "create" a particle at point x_1. He lets a source to disturb the homogeneous Dirac equation with a Dirac delta -shaped pulse. Then he tracks the time development of the wave function ψ. The Dirac equation contains the first time derivative of ψ.
This is not analogous to hitting a string with an infinitely narrow hammer. The hammer will give a momentum to an infinitely short segment of the string in an infinitely short time. The wave equation of a string contains the second time derivative of the vertical displacement of the string. The source in such an equation means the momentum of the string segment.
Electromagnetic waves are analogous to a string, but the Schrödinger equation is not.
We want to set ψ to some initial value at t = 0, then calculate its development forward in time.
If we want to do the calculation by summing the time development of small "spike" functions, we should set ψ = spike and calculate forward. We set a small spike as the initial value the wave function by setting a Dirac delta function (a "big" spike) as the source of our linear equation.
Basically, we can write
dψ / dt = Lψ(x, t) + source(x, t),
where source is a Dirac delta function in t and x, and zero elsewhere.
Thus, a short timestep forward:
ψ = ψ + Lψ dt + source dt.
When we loop the above algorithm for the very short time interval that source differs from 0, the equation above is dominated by the source and we can let it set the desired value as the initial value of the wave function.
Using the source method makes sense when some other field "disturbs" our ψ for a very short time in a small spatial zone and the disturbance can be modeled as a source function in our equation. But why would we use the source method to solve a simple initial value problem?
In Feynman diagrams, the most important thing is the vertexes where a field disturbs another field and creates particles to the other field. In those vertexes, it makes sense to use a Green's function, because it is a disturbance, that is, a source that creates particles.
Is it ok to use a Dirac delta function as the source, or should we use smoother functions?
A Dirac field might be seen as describing a point particle, the quantum of the field that has a sharp point location. The particle then disturbs the electromagnetic field and creates quantums.
In bremsstrahlung, the photons are created in the interaction of two charges. Does it make sense to put them in a Feynman diagrams as originating from a single charge?
What is the role of virtual particles in the Lagrangian and Feynman diagrams? The Feynman integrals are done in the "energy-momentum space". How do we describe a virtual particle there versus a real particle?
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