We may model breaking into more dimensions using potential walls of the following shape:
|
______|
o --->
______
|
|
The particle moves in a narrow corridor and comes to a wider area. In classical mechanics, there is no problem in such a process. But in quantum mechanics, the particle will have a huge vertical momentum. We knew the vertical position of the particle accurately when it was in the corridor. The uncertainty principle implies it had a huge momentum in the vertical direction.
How can we model breaking into more degrees of freedom, or into more dimensions, in quantum mechanics?
Maybe we should require that after a sudden freedom, the particle is again forced into a narrow corridor:
| |_______
_____| _______
_____ o--> |_______
Maybe we should require that after a sudden freedom, the particle is again forced into a narrow corridor:
| |_______
_____| _______
_____ o--> |_______
| ________
| |
| |
Feynman in his diagrams simply prunes away solutions where outgoing particles have more energy than incoming particles. Huge momenta are allowed in internal lines, but outgoing particles must be on-shell. It is like the corridor system above.
e- -------->
|
|
Z
Even the simplest Feynman diagram of a collision involves breaking into more degrees of freedom. The virtual photon adds more dimensions to the system.
We calculated that if the collision energy is a modest one electron mass, and we want to construct a wave packet whose size is the collision area 10^-15 m, then the packet will contain momenta which require up to 200 electron masses if we assume that the particle moves at almost the speed of light.
Wave mechanics requires high momenta to be able to model collisions. Breaking into more degrees of freedom is one aspect of this.
The rest mass of the electron is small compared to these momenta. In this blog we have written a lot about how a new pair has to tunnel through the potential barrier of creating two electron rest masses. But compared to the momenta involved, the barrier is very low.
Classically, pair production would be driven by the collision energy and the barrier of creating two electron rest masses. In wave mechanics, much bigger momenta come into the picture.
In wave mechanics, off-shell particles can use their huge momentum to tunnel through potential barriers in the collision area. But they must shed their huge momentum before leaving the collision area, so that they are on-shell.
In vacuum polarization, what is the role of the electric field in the collision area? Classically, it is the driver of polarization. In a Feynman diagram, a virtual photon breaks into more dimensions by forming a pair. The pair will have huge momenta. Then we should calculate the probability amplitude that the pair meets to annihilate. How is the electric field involved in the Feynman calculation? It is in the smallish momentum q which the photon was carrying. The big difference in the classical picture and the quantum mechanical picture is the huge momenta which the pair has.
Exponential attenuation of off-shell paths?
Suppose that a virtual photon is converted to a pair. We express the electron and the positron as wave packets. The packets contain plane waves with huge momenta.
Maybe we should let the probability amplitude of off-shell particles decrease exponentially with time? Such a wave describes an electron which has "borrowed energy from the vacuum". In the ordinary Schrödinger equation, a particle inside a potential barrier "has borrowed energy" and its wave function decays exponentially as we go deeper into the barrier.
In the Schrödinger case, the phase of the wave function is frozen inside the barrier. The phase there does revolve with time, but the phase is the same regardless of the spatial position.
In the Feynman diagram case, should we freeze the phase of off-shell particles?
An off-shell particle can "borrow energy from the vacuum", so that its four-momentum has E matching the p_x, p_y, p_z. A real particle has a plane wave function
exp(i p • (t, x) * 2π / h),
where we have just one spatial coordinate x.
We may imagine that an off-shell particle at the origin (0, 0) borrows the required ΔE of energy from a "brother" particle, whose kinetic energy after that is ΔE negative.
We imagine that the brother tunnels through a potential barrier and meets the original particle at x. Using non-relativistic physics,
We imagine that the brother tunnels through a potential barrier and meets the original particle at x. Using non-relativistic physics,
p_x = i sqrt(2m ΔE),
where p_x is the momentum of the brother and m is the rest mass of the brother.
The wave function of the brother decays as
exp(i p_x * x * 2π / h)
= exp(-x sqrt(2m ΔE) * 2π / h)
= exp(-x sqrt(2m ΔE) * 2π / h)
with x.
The Heisenberg uncertainty principle suggests that the decay should be something like
exp(-t *ΔE *4π / h).
Is there any formula for a relativistic Dirac particle tunneling through a potential barrier?
Let us assume that the decay derived from the Heisenberg formula is the right one.
We also assume that an off-shell particle can borrow momentum from the vacuum, so that it keeps its velocity vector after borrowing the energy to become real. For example, a positron with -511 kB of energy, moving to the positive z direction and the momentum p_z to the negative z direction, will keep moving to the positive z direction after taking the loan.
The Feynman integral for a vacuum polarization loop is roughly
p-slash - m -p-slash - m
∫ ---------------- • ------------------- dp^4 • dx^4
p^2 - m^2 p^2 - m^2
Above we assume that the momentum q of the virtual photon is negligible. The dx^4 is the integral of the product of the plane waves
exp(p • x) * exp(-p • x) = 1
over the whole timespace. Only pairs (p, -p) contribute considerably to the integral because otherwise the product of the plane waves rotates in phase and the contribution is "smallish".
The integral over dx^4 is infinite, of course. It has to be normalized somehow.
The p part of the integral diverges like p^2 because the integral over dp^4 brings a factor p^4 and it is only balanced by 1/p^2 in the formula.
Now if the "lifetime" of the particles goes like 1 / |p|, the integral over dx^4 will go like 1 / |p^4|, and the Feynman integral will converge.
But we have to think about pairs (p, p') where p' is not -p. The integral of the product of the plane waves is no longer over the whole timespace, but is constrained by the lifetime of the particle. Then pairs (p, -p) no longer dominate the integral. Can we enforce a rule that four-momentum has to be preserved in a vertex? Then we could restrict us to (p, -p) pairs.
Also, the whole Fourier decomposition might not make sense if the lifetime of the particle is limited. The Fourier decomposition is derived from the Dirac equation, where the particle is eternal.
Note that there is no infrared divergence for small |p|. That is because the virtual pair has to borrow from the vacuum at least 2 electron masses worth of energy. If the virtual photon has no energy of its own, the maximum lifetime of a virtual pair is roughly
Δt = h / (4π ΔE)
= 5 * 10^-35 / (10^6 * 1.6 * 10^-19)
= 3 * 10^-22 s,
and the pair can travel at most
3 * 10^8 * 3 * 10^-22
= 10^-13 m
in that time.
We can compare 10^-13 m to the Compton wavelength of the electron, 2 * 10^-12 m.
If we have a force carried by a massive particle, then a well-known formula
h / (2π m c)
gives its range. Let us calculate the range, assuming that m is two electron masses. We get 1.6 * 10^-13 m.
Proposition 1. We found a natural way to make the Feynman vacuum polarization integral convergent by assuming that the probability amplitude of an off-shell electron or positron decays as
exp(-t ΔE * 4π / h),
where ΔE is the energy needed to make the particle on-shell and t is the time. We measure both ΔE and t in the center of mass coordinates of the collision. QED.
The formula in Proposition 1 is not Lorentz invariant, because we measure t and ΔE in a preferred coordinate system, which is the center of mass of the collision. A more serious flaw is that it is not time symmetric: the wave function decays when we move "forward" in time in the laboratory frame.
Tunneling in general seems to be a time asymmetric. The system tends to move into a state of higher entropy. A collision of particles tends to increase entropy. The time asymmetry in Proposition 1 has its origin in this. How to make Proposition 1 time symmetric? We could have particles arriving from the future, whose wave function would decay as we move backward in time.
We need to think if Proposition 1 should be enforced on virtual photons, too. Photons never need to tunnel. They just experience a redshift. The electric force is a long-range force. An exponential decay of the probability amplitude does not fit well with a 1/r^2 force. Maybe Proposition 1 is only for particles with a non-zero rest mass?
What is the basic difference between the Feynman thinking and Proposition 1? Feynman seems to think that in the virtual pair the electron is a "real" particle and that it returns back to its original position traveling back in time as a positron. The propagator is symmetric in time, Lorentz invariant, and does not decay though the positron as seen by an observer with the usual direction of time is very much a virtual particle with a serious shortage of energy.
In the spirit of Proposition 1, we may view the electron as tunneling first forward in time, with a decaying wave function, and then tunneling back in time, as its wave function decays even more.
Is there something like tunneling forward and backward in time in the Schrödinger equation?
Does the phase of the wave function rotate in tunneling?
In the Schrödinger equation, the phase of the wave function is frozen for spatial distances while it does rotate in time. How is it in the Feynman diagrams? Should we freeze the phase spatially for a virtual pair or even the whole physical system?
Suppose that we have non-interacting systems A and B. Suppose that A is not tunneling. Then the phase of its wave function rotates normally. If B is tunneling, its phase is frozen spatially. It looks natural that we can take a product of the wave functions A and B to get a description of the combined system.
What about an interacting system? If particle A is tunneling through the potential imposed by particle B then the whole system can be described as one particle moving in a 7-dimensional timespace. In that case, does it make sense to freeze spatially the wave function of the whole system? Suppose that B is a much more massive particle than A. Why would an insignificant satellite A freeze the spatial wave function of the big B?
How would we model spatially rotating phase in a particle which is tunneling in the Schrödinger equation? Maybe the particle uses borrowed energy to overcome a 1.022 MeV barrier and then uses the rest of the borrowed energy as its kinetic energy? Then the phase of the particle will rotate as if it were a real particle flying over the barrier. The particle may borrow the energy from a "brother" which has to tunnel through a potential wall, causing the product of the wave functions to decay exponentially.
As a diagram:
O flying
---------------> additional kinetic E
_________
| | 1.022 MeV potential
O -----------> | | ---------> O
| |
B --------->------------------>---------> B
B tunneling
Above, the circle O represents the pair. B is the brother particle. B has no kinetic energy and lives on a potential V. There is a pit in the potential V. B falls into the pit (not depicted in the diagram) and bumps into pair O, giving the pair 1.022 MeV plus some additional kinetic energy.
The pair flies over the barrier. After the pit, B should return immediately to the normal level of V. But it does not have enough energy. It has to tunnel through V until it meets the pair O again and receives the borrowed energy back.
In the scheme above, the pair O behaves like a real pair flying over the potential barrier, adding cycles to it phase as it moves. The brother B has the hard task of tunneling. The wave function of B will decay exponentially as it tries to get the through and meet the pair O again and receive the energy that B gave out as a loan.
What did our diagram accomplish? It is an example of a Schrödinger equation process where a pair borrows energy to exist and move as a free pair, adding cycles to its wave function as it moves. The decaying wave function is the burden of the brother B which gave out the loan.
We conclude that the phase of the virtual pair can rotate spatially in a Feynman diagram, as if the pair were real. The exponentially decaying factor is a separate coefficient. In the above diagram we put the burden on a brother particle B. We may identify B as the "vacuum" from which a particle can borrow energy temporarily.
Suppose that we have non-interacting systems A and B. Suppose that A is not tunneling. Then the phase of its wave function rotates normally. If B is tunneling, its phase is frozen spatially. It looks natural that we can take a product of the wave functions A and B to get a description of the combined system.
What about an interacting system? If particle A is tunneling through the potential imposed by particle B then the whole system can be described as one particle moving in a 7-dimensional timespace. In that case, does it make sense to freeze spatially the wave function of the whole system? Suppose that B is a much more massive particle than A. Why would an insignificant satellite A freeze the spatial wave function of the big B?
How would we model spatially rotating phase in a particle which is tunneling in the Schrödinger equation? Maybe the particle uses borrowed energy to overcome a 1.022 MeV barrier and then uses the rest of the borrowed energy as its kinetic energy? Then the phase of the particle will rotate as if it were a real particle flying over the barrier. The particle may borrow the energy from a "brother" which has to tunnel through a potential wall, causing the product of the wave functions to decay exponentially.
As a diagram:
O flying
---------------> additional kinetic E
_________
| | 1.022 MeV potential
O -----------> | | ---------> O
| |
B --------->------------------>---------> B
B tunneling
Above, the circle O represents the pair. B is the brother particle. B has no kinetic energy and lives on a potential V. There is a pit in the potential V. B falls into the pit (not depicted in the diagram) and bumps into pair O, giving the pair 1.022 MeV plus some additional kinetic energy.
The pair flies over the barrier. After the pit, B should return immediately to the normal level of V. But it does not have enough energy. It has to tunnel through V until it meets the pair O again and receives the borrowed energy back.
In the scheme above, the pair O behaves like a real pair flying over the potential barrier, adding cycles to it phase as it moves. The brother B has the hard task of tunneling. The wave function of B will decay exponentially as it tries to get the through and meet the pair O again and receive the energy that B gave out as a loan.
What did our diagram accomplish? It is an example of a Schrödinger equation process where a pair borrows energy to exist and move as a free pair, adding cycles to its wave function as it moves. The decaying wave function is the burden of the brother B which gave out the loan.
We conclude that the phase of the virtual pair can rotate spatially in a Feynman diagram, as if the pair were real. The exponentially decaying factor is a separate coefficient. In the above diagram we put the burden on a brother particle B. We may identify B as the "vacuum" from which a particle can borrow energy temporarily.
Lorentz invariance in Proposition 1
Proposition 1 clearly breaks Lorentz invariance. Let us analyze why we are allowed to use a preferred coordinate system. The coordinate system might be the center of mass of the collision, or any inertial coordinate system which is fixed before the collision.
As we wrote in an earlier post, in a collision there is energy flow to/from/within the electric field regardless of the inertial coordinate system which we use to measure the flow. The virtual photon symbolizes this energy flow. Maybe the four-momentum q that it transfers is a good measure of the energy flow, in any inertial coordinate system?
Vacuum polarization tries to counter the energy flow. But how? In a subsequent blog post we need to analyze how exactly the Feynman model of a virtual pair affects the energy flow. The pair depends on q.
Real pair production reduces the energy flow, as we explained with a classical model in an earlier blog post:
http://meta-phys-thoughts.blogspot.com/2018/09/what-is-quantum-of-arbitrary-energy-flow.html
We need to analyze precisely what is the role of the virtual pairs.
When real pairs are formed in a collision, they steal some of the energy which would have flowed.
Maybe virtual pairs are like extra elasticity of the rod that pulls the colliding particles together? The Feynman model says that the electric force feels weaker at long distances. The rod would have more extra elasticity the longer it is.
Real pair production reduces the energy flow, as we explained with a classical model in an earlier blog post:
http://meta-phys-thoughts.blogspot.com/2018/09/what-is-quantum-of-arbitrary-energy-flow.html
We need to analyze precisely what is the role of the virtual pairs.
When real pairs are formed in a collision, they steal some of the energy which would have flowed.
Maybe virtual pairs are like extra elasticity of the rod that pulls the colliding particles together? The Feynman model says that the electric force feels weaker at long distances. The rod would have more extra elasticity the longer it is.
In earlier blog posts we tossed the idea that a system of particles flying in space "builds its own environment". Its neigborhood is filled with the static electric field of the particles. We cannot expect Lorentz invariance for phenomena which depend on these electric fields.
Kenneth Wilson studied cutoffs in the 1970s. We need to look at his papers.
Loop quantum gravity tries to discretize space. A main problem is Lorentz invariance. How to make the grid Lorentz invariant? Maybe it could be done by attaching the local grid to the center of mass of the local interacting system?
Running of the coupling constant
Proposition 1 makes the vacuum polarization integral to converge. Vacuum polarization counters the electric force between colliding particles. Does the measured force depend on the collision energy?
Experimental results from LEP suggest that the effective fine structure constant in a 100 GeV collision is 1/129, not 1/137 like in a low energy collision. The error margin from LEP is big, though.
The lifetime of the virtual pair in the vacuum polarization loop is higher if |p| is smaller. Above we calculated that the pair may travel at most 10^-13 m before annihilating. After that, the virtual photon has to reach the other colliding particle. What if the collision happened in a much smaller area? What is the probability amplitude of the p photon reaching the other colliding particle?
If common sense is right, then in a high-energy collision, the vacuum polarization integral is reduced because for small |p|, some of the volume integral dx^4 is cut off.
The electric force will appear stronger at short distances.
The "bare charge" of the electron is finite in the model of Proposition 1. The vacuum polarization integral converges and we did not need to refer to any energy scale. We should calculate the integral so that we can compare the apparent charge to the experimental result from LEP.
Regularization versus tunneling, a tachyon appears
In the link on page we have a simple regularization of the vacuum polarization integral.
Λ seems to be the rest mass of a fictitious heavy particle.
Λ^4
----------------
(Λ^2 - p^2)^2
is the term which makes the Feynman integral to converge.
Our tunneling method of Proposition 1 involves a term which is roughly
h^4
----------------------------
64 π^4 (2 * m_e * c + p)^4.
It makes the Feynman integral to converge. There m_e is the rest mass of the electron and c the speed of light. The 2 * m_e * c comes from the fact that the loaned energy in tunneling is at least two electron rest masses if the energy of the virtual photon is zero.
At high p, our term behaves like the regulator above. At low p, our term is a little like the regulator where Λ has an imaginary value. That is, there is a particle with an imaginary rest mass traveling with the pair. A particle with an imaginary rest mass is sometimes called a tachyon.
A plane wave
exp(i (t * mc^2 - mv x))
decays exponentially with time if m is imaginary. If we imagine that a tachyon is moving along with the pair, it will make the product of the wave function of the pair and the tachyon to decay exponentially with time. The role of v and x is not clear, though, in the plane wave case.
Alternatively, we may have m real but t imaginary. Then the plane wave decays as with a tachyon, when t increases on the imaginary coordinate axis.
A plane wave
exp(i (t * mc^2 - mv x))
decays exponentially with time if m is imaginary. If we imagine that a tachyon is moving along with the pair, it will make the product of the wave function of the pair and the tachyon to decay exponentially with time. The role of v and x is not clear, though, in the plane wave case.
Alternatively, we may have m real but t imaginary. Then the plane wave decays as with a tachyon, when t increases on the imaginary coordinate axis.
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