Symmetry, Transactions, and the Mechanism of Wave Function Collapse

Article

Symmetry, Transactions, and the Mechanism of Wave

Function Collapse

John G. Cramer

and Carver A. Mead

Department of Physics, University of Washington, Seattle, WA 98195; jcramer@uw.edu

California Institute of Technology, Pasadena, CA 91125; carver@caltech.edu

Correspondence: jcramer@uw.edu

Received: date; Accepted: date; Published: date

Abstract:

The Transactional Interpretation of quantum mechanics exploits the intrinsic time-symmetry

of wave mechanics to interpret the

and

* wave functions present in all wave mechanics calculations

as representing retarded and advanced waves moving in opposite time directions that form a quantum

"handshake" or transaction. This handshake is a standing-wave that builds up across space-time to

transfer the conserved quantities of energy, momentum, and angular momentum in an interaction. Here

we derive a two-atom quantum formalism describing a transaction. Using this, we perform calculations

employing the standard wave mechanics of Schrödinger to show the transfer of energy from a hydrogen

atom in an excited state to a nearby hydrogen atom in its ground state, using the retarded and advanced

electromagnetic four-potentials. It is seen that the initial exchange creates a dynamically unstable situation

that avalanches to the completed transaction, demonstrating that wave function collapse, considered

mysterious in the literature, can be implemented in the Schrödinger formalism by including advanced

potentials without the introduction of any additional mechanism or formalism.

Keywords:

quantum mechanics; transaction; transactional interpretation; wave function collapse;

collapse mechanism

1. Introduction

Quantum mechanics (QM) was never properly finished. Instead, it was left in an exceedingly

unsatisfactory state by its founders. Many attempts by highly qualified individuals to improve the

situation have failed to produce any consensus about either (a) the precise nature of the problem, or (b)

what a better form of QM might look like.

At the most basic level, a simple observation illustrates the central conceptual problem:

An excited atom somewhere in the universe transfers

all

of its excitation energy to another single atom,

independent of the presence of the vast number of alternative atoms that could have received all or part of

the energy. The obvious "photon-as-particle" interpretation of this situation has a one-way symmetry: The

excited source atom is depicted as emitting a particle, a

photon

of electromagnetic energy that is somehow

oscillating with angular frequency

while moving in a particular direction. The photon is depicted as

carrying a quantum of energy

, a momentum

, and an angular momentum

through space, until it

is later absorbed by some unexcited atom. The emission and absorption are treated as independent unitary

events without internal structure. It is insisted that the only real and meaningful quantities describing this

process are

probabilities

, since these are measurable. The necessarily abrupt change in the quantum wave

function of the system when the photon arrives (and an observer potentially gains information) is called

"wave function collapse" and is considered to be a mysterious process that the founders of QM found it

necessary to "put in by hand" without providing any mechanism.

Further, there is the unresolved problem of

entanglement

. Over the past few decades, many increasingly

exquisite "EPR" experiments have demonstrated that multi-body quantum systems with separated

arXiv:2006.11365v1 [quant-ph] 19 Jun 2020

2 of 34

components that are subject to conservation laws exhibit a property called "quantum entanglement"[

Their component wave functions are inextricably locked together, and they display a nonlocal correlated

behavior enforced over an arbitrary interval of space-time without any hint of an underlying mechanism

or any show of respect for our cherished classical "arrow of time." Entanglement is the most mysterious of

the many so-called "quantum mysteries."

It has thus become clear that the quantum transfer of energy must have quite a different symmetry

from that implied by this simple "photon-as-particle" interpretation. Within the framework of statistical

QM, the intrinsic symmetry of the energy transfer and the mechanisms behind wave function collapse

and entanglement have been greatly clarified by the

Transactional Interpretation of quantum mechanics

(TI), developed over several decades by one of us

and recently described in some detail in the book

The

Quantum Handshake

[1], of which this paper is, in part, a review.

1.1. Wheeler-Feynman Electrodynamics

The Transactional Interpretation was inspired by classical time-symmetric Wheeler-Feynman

electrodynamics[

] (WFE), sometimes called "absorber theory".

Basically, WFE assumes that

electrodynamics must be time-symmetric, with equally valid retarded waves (that arrive

after

they are

emitted) and advanced waves (that arrive

before

they are emitted). WFE describes a "handshake" process

accounting for emission recoil in which the emission of a retarded wave stimulates a future absorber to

produce an advanced wave that arrives back at the emitter at the instant of emission. WFE is based on

electrodynamic time symmetry but has been shown to be completely interchangeable with conventional

classical electrodynamics in its predictions.

WFE asserts that the breaking of the intrinsic time-symmetry to produce the electromagnetic arrow of

time, i.e., the observed dominance of retarded radiation and absence of advanced radiation in the universe,

arises from the presence of more absorption in the future than in the past. In an expanding universe, that

assertion is questionable. One of us has suggested an alternative cosmological explanation[

], which

employs advanced-wave termination and reflection from the singularity of the Big Bang.

1.2. The Transactional Interpretation of Quantum Mechanics

The Transactional Interpretation of quantum mechanics[

] takes the concept of a WFE handshake

from the classical regime into the quantum realm of photons and massive particles. The retarded and

advanced waves of WFE become the quantum wave functions

and

*. Note that the complex conjugation

* is in effect the application of the Wigner time-reversal operator, thus representing an advanced wave

function that carries negative energy from the present to the past.

Some critics of the Transactional Interpretation have questioned why it does not provide a detailed

mathematical description of how a transaction forms

. This question, of course, betrays a fundamental

misunderstanding of what an

interpretation

of quantum mechanics actually is. In our view, the mathematics

is (and should be) exclusively contained in the quantum mechanics formalism itself. The function of the

interpretation is to

interpret

that mathematics, not to introduce some additional variant formalism.

In the

We note that Ruth Kastner has extended her "probabilist" variant of the TI into the quantum-relativistic domain[

] and has

used it to extend and enhance the "decoherence" approach to quantum interpretation[4].

Here we have done just that, except not as an interpretation but as a calculation using the Schrödinger formalism.

We note, however, that this principle is violated by the Bohm-de Broglie "interpretation", by the Ghirardi-Rimini-Weber

"interpretation", and by many other so-called interpretations that take the questionable liberty of modifying the standard QM

formalism. In that sense, these are alternative variant quantum theories,

not

interpretations at all.

3 of 34

present work, we use the standard Schrödinger wave mechanics formalism

with the inclusion of retarded and

advanced electromagnetic four-potentials

to describe and illuminate the processes of transaction formation

and the collapse of the wave function.

The TI leans heavily on the standard formalism of Schrödinger wave mechanics. However, that

formalism is conventionally regarded as not containing any mathematics that explicitly accounts for wave

function collapse (which the TI interprets as

transaction formation

). Here we show that this is incorrect, and

that the Schrödinger formalism with the inclusion of retarded and advanced 4-potentials can provide a

detailed mathematical description of a "quantum-jump" in which what seems to be a photon is emitted

by one hydrogen atom in an excited state and excites another hydrogen atom initially in its ground state.

Thus, the mysterious process of wave function collapse becomes just a phenomenon involving an exchange

of waves that is actually a part of the Schrödinger formalism.

As illustrated in Fig. 1, the process described involves the initial existence in each atom of a very

small admixture of the wave function for the opposite state, thereby forming two-component states in

both atoms. This causes them to become weak dipole radiators oscillating at the same difference frequency

. The interaction that follows, characterized by a retarded-advanced exchange of 4-vector potentials,

leads to an exponential build-up of a transaction, resulting in the complete transfer of one photon-worth

of energy ̄

from one atom to the other. This process is described in more detail below.

Figure 1.

Model of transaction formation: An emitter atom

2 in a space-antisymmertric excited state

of energy

and an absorber atom

1 in a space-symmetric ground state of energy

both have slight

admixtures of the other state, giving both atoms dipole moments that oscillate with the same difference

frequency

−

. If the relative phase of the initial small offer

and confirmation

∗

waves is

optimal, this condition initiates energy transfer, which avalanches to complete transfer of one photon-worth

of energy ̄

2. Physical Mechanism of the Transfer

The standard formalism of QM consists of a set of arbitrary rules, conventionally viewed as dealing

only with probabilities. When illuminated by the TI, that formalism hints at an underlying physical

mechanism that might be understood, in the usual sense of the concept

understood

. The first glimpse of

such an understanding, and of the physical nature of the transactional symmetry, was suggested by Gilbert

N. Lewis in 1926[8], the same year he gave the energy transfer the name "photon":

4 of 34

It is generally assumed that a radiating body emits light in every direction, quite regardless of whether

there are near or distant objects which may ultimately absorb that light; in other words that it radiates

"into space"...

I am going to make the contrary assumption that an atom never emits light except to another atom...

I propose to eliminate the idea of mere emission of light and substitute the idea of transmission, or a process

of exchange of energy between two definite atoms... Both atoms must play coordinate and symmetrical

parts in the process of exchange...

In a pure geometry it would surprise us to find that a true theorem becomes false when the page upon

which the figure is drawn is turned upside down. A dissymmetry alien to the pure geometry of relativity

has been introduced by our notion of causality.

In what follows we demonstrate that the pair of coupled Schrödinger equations describing the two

atoms, as coupled by a relativistically correct description of the electromagnetic field, exhibit a unique

solution. This solution has exactly the symmetry of the TI and thus provides a

physically understandable

mechanism for the experimentally observed behavior: Both atoms, in the words of Lewis, "

play coordinate

and symmetrical parts in the process of exchange.

The solution gives a smooth transition in each of the atomic wave functions, brought to abrupt closure

by the highly nonlinear increase in coupling as the transition proceeds. The origin of statistical behavior

and "quantum randomness" can be understood in terms of the random distribution of wave-function

amplitudes and phases provided by the perturbations of the many other potential recipient atoms; no

"hidden variables" are required. These findings might be viewed as a first step towards a physical

understanding of the process of quantum energy transfer.

We will close by indicating the deep, fundamental questions that we have not addressed, and that

must be understood before anything like a complete physical understanding of QM is in hand.

3. Quantum States

In 1926, Schrödinger, seeking a wave-equation description of a quantum system with mass, adopted

Planck’s notion that energy was somehow proportional to frequency together with deBroglie’s idea that

momentum was the propagation vector of a wave and crafted his wave equation for the time evolution of

the wave function

[9]:

−

m i

∇

∂

(1)

Here

is the electrical potential and

is the (negative) charge on the electron. So what is the meaning of

the wave function

that is being characterized? In modern treatments

is called a "probability amplitude",

which has only a probabilistic interpretation. In what follows, however, we return to Schrödinger ’s original

vision, which provides a detailed physical picture of the wave function and how it interacts with other

charges:

The hypothesis that we have to admit is very simple, namely that the square of the absolute value of

is proportional to an electric density, which causes emission of light according to the laws of ordinary

electrodynamics.

So we can visualize the electron as having a smooth charge distribution in 3-dimensional space,

whose density is given by

∗

. There is no need for statistics at any point of the calculation, and none of

the quantities have statistical meaning. The probabilistic outcome of quantum experiments has the same

origin as it does in all other experiments—random perturbations beyond the control of the experimenter.

We return to the topic of probability after we have established the nature of the transaction.

5 of 34

For a local region of positive potential

, for example near a positive proton, the negative electron’s

wave function has a local potential energy (

) minimum in which the electron’s wave function can form

local

bound states

. The spatial shape of the wave function amplitude is a tradeoff between getting close to

the proton, which lowers its potential energy, and bunching together too much, which increases its

∇

"kinetic energy." Eq. 1 is simply a mathematical expression of this tradeoff. A discrete set of states called

eigenstates

are standing-wave solutions of Eq. 1 of the form

−

, where

and

are functions of

only the spatial coordinates, and the angular frequency

is itself not a function of time. For the hydrogen

atom, the potential

, where

is the positive charge on the nucleus. Two of the lowest-energy

solutions to Eq. 1 with this potential are:

100

−

√

−

210

r e

−

cos

(

)

√

−

(2)

where the dimensionless radial coordinate

is the radial distance divided by the

Bohr radius

πe

0.0529 nm,

(3)

and

is the azimuthal angle from the North Pole (

axis) of the spherical coordinate system.

The spatial "shape" of the two lowest energy eigenstates for the hydrogen atom is shown in Fig. 2. Here we

focus on the excited-state wave function

210

that has no angular momentum projection on the z-axis. For

the moment, we set aside the wave functions

, which have

1 and

−

1 angular momentum z-axis

projections. Because, for any single eigenstate, the electron density is

∗

−

, the charge

density is not a function of time, so none of the other properties of the wave function change with time.

The individual eigenstates are thus

stationary states

. The lowest energy bound eigenstate for a given form

of potential minimum is called its

ground state

, shown on the left in Fig. 3. The corresponding charge

densities are shown in Fig. 4.

Already in 1926 Schrödinger had derived the energies and wave functions for the stationary solutions

of his equation for the hydrogen atom.

His physical insight, that the absolute square

∗

of the wave function was the

electron density

, had

enabled him to work out the energy shifts of these levels caused by external applied electric and magnetic

fields, the expected strengths of the transitions between pairs of energy levels, and the polarization of light

from certain transitions. These predictions could be compared directly with experimental data, which had

been previously observed but not understood.

He reported that these calculations were:

...not at all difficult, but very tedious. In spite of their tediousness, it is rather fascinating to see all the

well-known but not understood "rules" come out one after the other as the result of familiar elementary

and absolutely cogent analysis, like e.g. the fact that

∫

cos

vanishes unless

Once the hypothesis about

∗

has been made, no accessory hypothesis is needed or is possible; none

could help us if the "rules" did not come out correctly. But fortunately they do[9,10].

Schrödinger’s approach allows us to describe continuous quantum transitions in a simple and

intuitively appealing way by extending the notions of Collective Electrodynamics[

] to the wave function

of a single electron. We shall require only the most rudimentary techniques of Schrödinger’s original

quantum theory.

6 of 34

Figure 2.

Angular dependence of the spatial wave function amplitudes for the lowest (100, left) and

next higher (210, right) states of the hydrogen atom, plotted as unit radius in spherical coordinates from

Eq. 2. The 100 wave function has spherical symmetry: Positive in all directions. The 210 wave function is

antisymmetric along the

axis, as indicated by the color change. In practice the direction of the

axis will

be established by an external electromagnetic field, as we shall analyze shortly.

-

-

0.0

0.1

0.2

0.3

0.4

0.5

z

-

-

-

0.05

0.00

0.05

z

Figure 3.

Wave function amplitudes for the 100 and 210 states, along the

axis of the hydrogen atom. The

horizontal axis in all plots is the position along the

axis in units of the Bohr radius.

4. The Two-State System

Let us first consider a simple two-state system. The system has a single positive charge distribution

around which there are two eigenstates, labeled 1 and 2, that an electron can occupy. In State 1, the electron

has wave function

and energy

; in State 2, it has wave function

and energy

−

(4)

where

/ ̄

and

/ ̄

We visualize the wave function as a representation of a totally continuous matter wave, and take

and

as real functions of the space coordinates. We can interpret all of the usual operations involving the

7 of 34

-

-

0.00

0.05

0.10

0.15

0.20

0.25

z

-

-

0.00

0.01

0.02

0.03

0.04

0.05

0.06

z

Figure 4.

Contribution of

−

"slices" at position

of wave function density

∗

to the total charge or

mass of the 100 and 210 states of the hydrogen atom. Both curves integrate to 1.

wave function as methods for computing properties of this continuous distribution. The only particularly

quantal assumption involved is that the wave function obeys a

normalization condition

∫

∗

vol

(5)

where the integral is taken over the entire three-dimensional volume where

is non-vanishing

This condition assures that the total charge will be a single electronic charge, and the total mass will

be a single electronic mass.

By construction, the eigenstates of the Schrödinger equation are real and orthogonal:

∫

vol

(6)

The first moment

〈

〉

of the electron distribution

along the atom’s

axis is:

〈

〉

∫

∗

vol,

(7)

where the integral is taken over all space where the wave function is non-vanishing.

Envelope functions like

and

generally die out exponentially with distance sufficiently far from the "region of interest,"

such as an atom. Integrals like this one and those that follow are taken out far enough that the part being neglected is within the

tolerance of the calculation.

In statistical treatments

〈

〉

would be called the "expectation value of

", whereas for our continuous distribution it is called

the "average value of

" or the "first moment of z." The electron wave function is a

wave packet

and is subject to all the Fourier

properties of one, as treated at some length in ref.[

]. For example a packet only 100 wavelengths in duration has a spread in

frequency (and therefore energy) of order 1%, resulting, for example, in the observed

natural line width

of radiation from an

atom. Similarly, a packet confined within 100 wavelengths in dimension has a spread in wave vector (and therefore momentum)

of order 1%. Because statistical QM insisted that electrons were "point particles", one was no longer able to visualize how

they could exhibit interference or other wave properties, so a set of rules was constructed to make the outcomes of statistical

experiments come out right. Among these was the

uncertainty principle

, which simply restated the Fourier properties of an

object described by waves in a statistical context. No statistical attributes are attached to any properties of the wave function in

this treatment.

8 of 34

This moment gives the position of the center of negative charge of the electron wave function relative

to the positive charge on the nucleus. When multiplied by the electronic charge

, it is called

the electric

dipole moment q

〈

〉

of the charge distribution of the atom:

〈

〉

∫

∗

vol.

(8)

From Eq. 7 and Eq. 4, the dipole moment for the

th eigenstate is:

〈

〉

∫

∗

vol

∫

∗

z R

vol

∫

z d

vol.

(9)

Pure eigenstates of the system will have a definite parity, i.e., they will have wave functions with

either even symmetry [

(

) =

(

−

)

], or odd symmetry [

(

) =

−

(

−

)

]. For either symmetry the

integral over

vanishes, and the dipole moment is zero. We note that even if the wave function did not

have even or odd symmetry, the dipole moment, and all higher moments as well, would be independent of

time. By their very nature, eigenstates are stationary states and can be visualized as standing-waves—none

of their physical spatial attributes can be functions of time.

The notion of stationarity is the quantum answer to the original question about atoms with electrons

orbiting a central nucleus:

Why doesn’t the electron orbiting the nucleus radiate its energy away?

In his 1917 book,

The Electron

, R.A. Millikan

anticipates the solution in his comment about the

. . . apparent contradiction involved in the non-radiating electronic orbit—a contradiction which would

disappear, however, if the negative electron itself, when inside the atom, were a ring of some sort, capable

of expanding to various radii, and capable, only when freed from the atom, of assuming essentially the

properties of a point charge.

Ten years before the statistical quantum theory was put in place, Millikan had clearly seen that a continuous,

symmetric electronic charge distribution would not radiate, and that the real problem was the assumption

of a point charge. The continuous wave nature of the electron implies a continuous charge distribution.

That smooth charge distribution can propagate around the nucleus and thereby generate a magnetic

moment, as observed in many atoms. The smooth propagation around the nucleus does not imply

radiation.

5. Transitions

In order to radiate electromagnetic energy, the charge distribution

must change with time

. Because the

spatial attributes of a system in a pure eigenstate are stationary, the system in such an eigenstate cannot

radiate energy. Because the eigenstates of the system form a complete basis set, any behavior of the system

can be expressed by forming a linear combination (superposition) of eigenstates. The general form of such

a superposition of two eigenstates is:

−

(10)

See [

]. Millikan was the first researcher to directly observe and measure the quantized charge on the electron with his famous

oil-drop experiment, for which he later received the Nobel prize. This reference contains a fascinating discussion of these

experiments, and a wonderful section contrasting the corpuscular and the ether theories of radiation.

9 of 34

where

and

are real amplitudes, and

and

are real constants that determine the phases of the

oscillations

and

Using

−

and

−

, the charge density

of the two-component-state wave function

is:

∗

(

−

)(

−

)

(

−

)

(

−

)

−

(

−

)

(

−

)

abR

cos

(

)

(11)

So the charge density of the two-component wave function is made up of the charge densities of the two

separate wave functions, shown in Fig. 4, plus a term proportional to the product of the two wave function

amplitudes. It reduces to the individual charge density of the ground state when

0 and to that of

the excited state when

1. The product term, shown in green in Fig. 5, is the only non-stationary

term; it oscillates at the transition frequency

. The integral of the total density shown in the right-hand

plot is equal to 1 for any phase of the cosine term, since there is only one electron in this two-component

state.

-

-

-

-

0.10

-

0.05

0.00

0.05

0.10

0.15

0.20

0.25

z

-

-

-

-

0.10

-

0.05

0.00

0.05

0.10

0.15

0.20

0.25

z

Figure 5.

Left:

Plot of the three terms in the wave-function density in Eq. 11 for an equal

(

√

)

superposition of the ground state (

, blue) and first excited state (

, red) of the hydrogen atom. The green

curve is a snapshot of the time-dependent

product term, which oscillates at difference frequency

Right:

Snapshot of the total charge density, which is the sum of the three curves in the left plot. The

magnitudes plotted are the contribution to the total charge in an

−

"slice" of

∗

at the particular

coordinate. All plots are shown for the phase such that

cos

(

) =

1. The horizontal axis in each

plot is the spatial coordinate along the

axis of the atom, given in units of the Bohr radius

. Animation

here[32]

All the

∗

plots represent the density of negative charge of the electron. The atom as a whole is

neutral because of the equal positive charge on the nucleus. The dipole is formed when the center of

10 of 34

charge of the electron wave function is displaced from the nucleus.

The two-component wave function must be normalized, since it is the state of one electron:

∫

∗

vol

∫

(

−

)

(

−

)

vol

∫

vol

∫

vol

(

−

)

∫

vol

(12)

Recognizing from Eq. 5 and Eq. 6 that the individual eigenfunctions are normalized and orthogonal:

∫

vol

∫

vol

∫

vol

(13)

Eq. 12 becomes

∫

∗

vol

(14)

represents the fraction of the two-component wave function made up of the lower state

, and

represents the fraction made up of the upper state

. The total energy

of a system whose wave function

is a superposition of two eigenstates is:

(15)

Using the normalization condition

1 and solving Eq. 15 for

, we obtain:

−

(16)

In other words,

is just the energy of the wave function, normalized to the transition energy, and using

as its reference energy. Taking

as our zero of energy and

−

, Eq. 16 becomes:

⇒

∂

(

)

∂

(17)

Using:

∫

〈

〉

max

(18)

the dipole moment of such a superposition can, from Eq. 11, be written:

dipole moment

〈

〉

cos

(

)

(19)

The factor

is called the

dipole matrix element

for the transition; it is a measure of the maximum

strength of the oscillating dipole moment. If one

is an even function of

and the other is an odd function

, as in the case of the 100 and 210 states of the hydrogen atom, then this factor is nonzero, and the

transition is said to be

electric dipole allowed

. If both

and

are either even or odd functions of

then this factor is zero, and the transition is said to be

electric dipole forbidden

. Even in this case, some

other moment of the distribution generally will be nonzero, and the transition can proceed by magnetic

11 of 34

dipole, magnetic quadrupole, or other higher-order moments. For now, we will concentrate on transitions

that are electric dipole allowed.

We have the time dependence of the electron dipole moment

〈

〉

from Eq. 19, from which we can

derive the velocity and acceleration of the charge:

〈

〉

cos

(

)

∂

〈

〉

∂

−

sin

(

) +

cos

(

)

∂

(

)

∂

≈−

sin

(

)

∂

〈

〉

∂

≈

cos

(

)

(20)

where the approximation arises because we will only consider situations where the coefficients

and

change slowly with time over a large number of cycles of the transition frequency:

(

∂

(

)

∂

)

The motion of the electron mass density endows the electron with a momentum

⇒

∂

〈

〉

∂

≈−

sin

(

)

(21)

6. Atom in an Applied Field

Schrödinger had a detailed physical picture of the wave function, and he gave an elegant derivation

of the process underlying the change of atomic state mediated by electromagnetic coupling.

Instead of directly tackling the transfer of energy between two atoms, he considered the response of a

single atom to a small externally applied vector potential field

. He found that the immediate effect of an

applied vector potential is to change the momentum

of the electron wave function:

∂

〈

〉

∂

−

q A

∂

〈

〉

∂

−

∂

(22)

So the quantity

−

∂

acts as a

force

, causing an

acceleration

of the electron wave function.

This is the physical reason that

−

∂

can be treated as an

electric field

−

∂

(23)

In the simplest case, the

q A

term makes only a tiny perturbation to the momentum over a single cycle of

the

oscillation, so its effects will only be appreciable over many cycles.

The original derivation[

] p.137, is not nearly as readable as that in Schrödinger ’s second and third 1928 lectures[

], where the

state transition is described in section 9 starting at the bottom of page 31, for which the second lecture is preparatory. There he

solved the problem more generally, including the effect of a slight detuning of the field frequency from the atom’s transition

frequency.

Far from an overall charge-neutral charge distribution like an atom, the longitudinal gradient of the scalar potential just cancels

the longitudinal component of

∂

, so what is left is

−

∂

⊥

∂

, which is purely transverse.

12 of 34

We consider an additional simplification, where the frequency of the applied field is exactly equal to the

transition frequency

of the atom:

cos

(

)

⇒ −

∂

sin

(

)

(24)

In such evaluations we need to be very careful to identify

exactly which energy

we are calculating:

The electric field is merely a bookkeeping device to keep track of the energy that an electron in one atom

exchanges with another electron in another atom, in such a way that the total energy is conserved. We will

evaluate how much energy a given electron gains from or loses to the field, recognizing that the negative

of that energy represents work done by the electron on the source electron responsible for the field. The

force on the electron is just

. Because

sin

(

)

, for a stationary charge, the force is in the

direction as much as it is in the

−

direction, and, averaged over one cycle of the electric field, the work

averages to zero. However, if the charge itself oscillates with the electric field, it will gain energy

∆

from

the work done by the field on the electron over one cycle:

∆

cycle

∫

∂

〈

〉

∂

(25)

where

〈

〉

is the

position of the electron center of charge from Eq. 19.

When the electron is "coasting downhill"

with

the electric field, it gains energy and

∆

is positive. When

the electron is moving "uphill"

against

the electric field, the electron loses energy and

∆

is negative.

As long as the energy gained or lost in each cycle is small compared with

, we can define a

continuous

power

(rate of change of electron energy), which is understood to be an average over many

cycles. The time required for one cycle is

, so Eq. 25 becomes:

∂

∆

∫

∂

〈

〉

∂

∫

∂

〈

〉

∂

(

)

(26)

7. Electromagnetic Coupling

Because our use of electromagnetism is conceptually quite different from that in traditional Maxwell

treatments, we review here the foundations of that discipline from the present perspective.

The entire

content of Maxwell’s Equations is contained in the relativistically correct Riemann–Sommerfeld differential

form:

(

∇

−

∂

)

−

(27)

where

= [

]

is the four-potential and

= [

]

is the four-current,

is the vector potential,

is the

scalar potential,

is the physical current density (no displacement current) and

is the physical charge

density, all expressed in the same inertial frame.

Connection with the usual electric and magnetic field quantities

and

is as follows:

−∇

−

∂

∇×

(28)

A more detailed discussion from the present viewpoint is given in reference[11].

The standard treatment is given in Jackson Chapter 8[15].

13 of 34

So, once we have the four-potential

we can derive any electromagnetic coupling we wish.

Eq. 27 has a completely general Green’s Function solution for the four-potential

(

)

at a point in space,

from four-current density

(

)

in volume element

vol at at distance

from that point:

(

) =

∫

(

)

∣

vol,

(29)

where

is the distance from element

vol to the point where

is evaluated.

The second-order nature of derivatives in Eq. 27 do not favor any particular sign of space or time. Eq. 29

can be expressed in terms of more familiar E&M quantities:

(

) =

∫

(

)

∣

vol

(

) =

∫

(

)

∣

vol.

(30)

If the charge density occurs as a small, unified "cloud", as is the case for the wave function of an atomic

electron, and the motion of the wave function is non-relativistic, the

integral just becomes the movement

of the center of charge:

(

)

≈

∫

(

)

∣

vol

≈

∣

(31)

Then, from a distance large compared to the size of the wave function,

can be taken from the time-average

center of charge, and, as we have chosen previously, the motion is in the

direction:

≈

∂

〈

〉

∂

∣

(32)

If, in addition, the time-average center of charge is stationary relative to the point where

is measured:

≈

∂

〈

〉

∂

∣

(33)

It is this form that we shall use for the simple examples presented.

An important difference between standard Maxwell E&M practice and our use of the of the

four-potential to couple atomic wave functions is highlighted by Wheeler and Feynman[5]:

There is no such concept as "the" field,

an independent entity with degrees of freedom of its own.

The field is simply a convenient bookkeeping device for keeping track of the total effect of an arbitrary

number of charges on a particular charge at some position in space. The fact that the field is "radiating into

space" does not imply that it is carrying energy with it. Energy is only transferred at the position of another

charge. Since all charges are the finite charge densities of wave functions, there are no self-energy infinities

in this formulation.

The assumption of point-charges has created untold conceptual complications, as

noted above in our discussion of stationary states.

Richard Feynman is famously known[

] to have abandoned his own WFE because the published version made the assumption

(false and unnecessary[

]) that an electric charge does not interact with the field it produces. That assumption was made to

eliminate the self-energy infinities of point-like electrons. However, Feynman later realized that some self-energy was needed to

explain the Lamb shift. Note that no such infinities exist in the present approach because the electron wave function cannot

contract to a point.

14 of 34

Although no energy is carried by "the" field, because it has no degrees of freedom of its own, electrical

engineers have developed extremely clever methods for using it as a bookkeeping method for keeping

track of the energy passing between distant charge distributions by pretending that the energy is carried

in the field. One of the most ingenious is the

Poynting Vector

, signifying the flow of power (energy per

unit time) per unit area:

E×

−

∂

(

∇×

)

(34)

We take as an example a z-polarized plane wave of amplitude

propagating in the

direction:

{

0, 0, cos

(

−

)

}

−

∂

{

0, 0,

−

sin

(

−

)

}

∇×

{

sin

(

−

)

, 0

}

E×

{

sin

(

−

)

, 0, 0

}

avg

{

, 0, 0

}

{

, 0, 0

}

(35)

where the braces signify

{

}

vectors, and the final form recognizes that, in free space,

We have ascertained how an atom in a superposed state of two eigenstates can gain or lose energy

from/to an external time-varying vector potential. We are naturally led to ask how much energy an atom

in such a state would radiate into "free space." Far from an overall charge-neutral charge distribution like

an atom, the longitudinal gradient of the scalar potential just cancels the longitudinal component of

∂

so what is left of the propagating wave is the purely transverse component of the vector potential. Since

the current in the atom is in the

direction, the vector potential will be in the

direction, but the surviving

propagating field at large distance will be only the transverse component

⊥

. In a spherical coordinate

system the transverse component of

will be

⊥

sin

(

)

. At large distances, all propagating waves

approach the nature of a plane wave, so we can use Eq. 35 in the form:

out

≈

⊥

sin

(

)

(36)

where

out

represents the outward-directed energy flow, whose units look like they are watts per m

. But

we must be careful—because of our plane-wave characterization it is only the power propagating in one

direction. So people in the business often give this quantity, called the

radiance

, whose units are watts per

per steradian. In our simple configuration the atom can be treated as effectively a point source, so the

total power

radiated will be just the integral of

out

over a spherical surface of radius

≈

∫

sin

(

)

sin

(

)

(37)

15 of 34

From Eq. 20 and Eq. 32, we obtain the vector-potential

at a distance

from the oscillating charge

distribution of our atom, neglecting phase and time delay:

∂

〈

〉

∂

−

a b

sin

(

)

≈

∂

〈

〉

∂

−

a b

sin

(

)

≈

sin

(

)

⇒

avg

≈

rad

where

rad

(38)

From the green curve in Fig. 5 we can estimate the dipole matrix element

, which is

times the "length"

between the positive and negative "charge lumps", say

≈

. So, for the H atom,

rad

≈

−

watt

Then from Eq. 17 we find the time course of the radiation process:

⇒

∂

−

rad

(

−

)

∂

(

)

∂

where

rad

≈

7.9

−

sec.

(39)

In the following two sections we will find that atoms spaced by an arbitrary distance exhibit transactions

of exactly the same form as shown in Fig. 6.

The frequency for this transition is

≈

1.55

, so in time

there are

≈

cycles.

This value justifies our assumption that

and

change slowly on the time scale of

The transition of an excited atom into "empty space" is called

spontaneous emission

, and was the

subject of considerable debate during the early development of QM. It was originally thought that the

spontaneous emission lifetime

, observed as the sharpness of the spectral line, was a local property of each

particular electronic transition. However in 1985 it was observed[

] that the lifetime was not fixed and

could be made much longer if the transition occurred in a waveguide beyond cutoff, thereby showing that

the energy from the transition had been propagating outward. This is now widely ascribed to the coupling

of the source electron to degrees of freedom of the radiation field. One widely-held viewpoint treats

the "quantum vacuum" as being made up of an infinite number of quantum harmonic oscillators. The

problem with this view is that each such oscillator would have a zero-point energy that would contribute

to the energy density of space in any gravitational treatment of cosmology. Even when the energies of the

oscillators are cut off at some high value, the contribution of this "dark energy" is 120 orders of magnitude

larger than that needed to agree with astrophysical observations. Such a disagreement between theory

and observation (called the "cosmological constant problem"), even after numerous attempts to reduce it,

many orders of magnitude worse than any other theory-vs-observation discrepancy in the history of science

The TI does not suffer from this serious defect, since its vacuum has no degrees of freedom of its own.

Where, then, is the radiated energy going? The obvious candidate is the enormous continuum of states

of matter in the early universe, source of the

cosmic microwave background

, to which atoms here and

now are coupled by the quantum handshake. The Poynting vector is just a bookkeeping mechanism for

summing up all the full or partial transitions to matter in that continuum. For independent discussions

from the two of us, see ref.[11] p.94 and ref.[7].

8. Two Coupled Atoms

The central point of this paper is to understand the

photon

mechanism by which energy is transferred

from a

single

excited atom (atom

) to another

single

atom (atom

) initially in its ground state. We proceed

16 of 34

with the simplest and most idealized case of two identical atoms, where:

(1) Excited atom

will start in a state where

≈

1 and

is very small, but never zero because of its

ever-present random statistical interactions with a vast number of other atoms in the universe, and

(2) Likewise, atom

will start in a state where

≈

1 and

is very small, but never zero for the same

reason.

Thus each atom starts in a two-component state that has an oscillating electrical current described by

Eq. 20:

∂

〈

〉

∂

≈−

sin

(

)

∂

〈

〉

∂

≈−

sin

(

)

(40)

where we have taken excited atom

as our reference for the phase of the oscillations (

0),

and the approximation assumes that

and

are changing slowly on the scale of

Although that random starting point will contain small excitations of a wide range of phases,

we simplify the problem by assuming the following:

All

of the electric field

at atom

is supplied by atom

All

of the electric field

at atom

is supplied by atom

The dipole moments of both atoms are in the

direction,

The atoms are separated by a distance

in a direction orthogonal to

The vector potential at distance

from a small charge distribution oscillating in the

-direction is,

from Eq. 33:

∂

〈

〉

∂

∣

(41)

Since all motions and fields are in the

direction, we can henceforth omit the

subscript.

When the distance

is small compared with the wavelength

the delay

can be neglected

Using Eq. 23 and Eq. 33, the vector potentials and electric fields from the two atoms become:

≈

∂

〈

〉

∂

⇒ E

−

∂

≈−

∂

〈

〉

∂

≈

∂

〈

〉

∂

⇒ E

−

∂

≈−

∂

〈

〉

∂

(42)

When atom

is subject to electric field

and atom

is subject to electric field

, the energy of both

atoms will change with time in such a way that the total energy is conserved. Thus the superposition

amplitudes

and

of both atoms change with time, as described by Eq. 17 and Eq. 26, from which:

∂

≈

∫

∂

〈

〉

∂

(

) =

−

∫

∂

〈

〉

∂

〈

〉

∂

(

)

∂

≈

∫

∂

〈

〉

∂

(

) =

−

∫

∂

〈

〉

∂

〈

〉

∂

(

)

(43)

See footnote for Eq. 23. Since this dimension is of the order of

−

m and the wavelength is of the order of

−

m, this case can

be accommodated. We shall find that the results we arrive at here are directly adaptable to the centrally important case in which

the atoms are separated by an arbitrarily distance, which will be analyzed in the next section.

17 of 34

From Eq. 43, using the

〈

〉

derivatives from Eq. 20:

∂

≈−

∫

(

−

cos

(

)

)(

−

sin

(

)

(

)

≈−

∫

cos

(

)

sin

(

)

(

) =

sin

(

)

∂

≈−

∫

(

−

cos

(

)

)(

−

sin

(

)

(

)

≈−

∫

cos

(

)

sin

(

)

(

) =

−

sin

(

)

(44)

These equations describe energy transfer between the two atoms in either direction, depending on the

sign of

sin

(

)

. For transfer from atom

to atom

∂

is negative. Since this transaction dominated all

competing potential transfers, its amplitude will be maximum, so

sin

(

) =

−

1. If the starting state had

been atom

in the excited (

≈

1) state, the sin

(

) = +

1 choice would have been appropriate.

Using :

sin

(

) =

−

and

αβ

(45)

Eq. 44 becomes:

∂

−

αβ

∂

αβ

(46)

Energy is conserved by the two atoms during the transfer, and the wave functions are normalized:

(

)

⇒

√

−

⇒

√

−

⇒

√

−

(47)

after which substitutions Eq. 46 becomes:

∂

(

)

∂

g b

(

−

)

where

αβ

(48)

which has solution:

−

(49)

The direction and magnitude of the entire energy-transfer process is governed by the relative phase

of the electric field and the electron motion in

both atoms

: When the electron motion of

either atom

in phase

with the field, the field transfers energy to the electron, and the field is said to

excite

the atom.

When the the electron motion has

opposite phase

from the field, the electron transfers energy "to the field",

and the process is called

stimulated emission

Therefore, for the photon transaction to proceed the field from atom

must have a phase such that it

"excites" atom

, while the field from atom

must have a phase such that is absorbs energy and "de-excites"

atom

. In the above example, that unique combination occurs when sin

(

) =

−

18 of 34

-

-

-

0.0

0.2

0.4

0.6

0.8

1.0

Figure 6.

Squared state amplitudes for atom

(blue) and

(red) for the Photon transfer of energy

from atom

to atom

, from Eq. 49. Using the lower state energy as the zero reference,

the energy of the state. The green curve shows the normalized power radiated by the atom

and absorbed

by atom

, from Eq. 48. The optical oscillations at

are not shown, as they are normally many orders of

magnitude faster than the transfer time scale

. In the next section we will find that atoms spaced by an

arbitrary distance exhibit transactions of exactly the same form.

The random starting point for the transaction involving one excited atom will contain small excitations

of a wide range of phases. From Eq. 48, the amplitude of each of those excitations will grow at a rate

proportional to its own phase match. Thus the excitation from a random recipient atom that happens to

have

sin

(

)

≈−

1 will win in the race and become the dominant partner in the coordinated oscillation of

both atoms.

Thus, we have identified the source of the intrinsic randomness within quantum mechanics, an aspect

of the theory that has been considered mysterious since its inception in the 1920s.

Each wave function will thus evolve its motion to follow the applied field to its maximum resonant

coupling, and we can take

sin

(

) =

−

1 in these expressions, which we have done in Eq. 46 and Fig. 6.

From a TI point of view, both atoms start in stable states, with each having extremely small admixtures

of the other state, so that both have very small dipole moments oscillating with angular frequency

= (

−

)

/ ̄

. We assume that in atom

this admixture creates an initial positive-energy offer wave

that interacts with the small dipole moment of absorber atom

to transfer positive energy, and that in

atom

this admixture creates an initial negative-energy confirmation wave from the excited emitter atom

that interacts with the small dipole moment of emitter atom

to transfer negative energy, as shown

schematically in Fig. 1. Because of these admixtures, both atoms develop small time-dependent dipole

moments that, because of the mixed-energy superposition of states as shown in Fig. 5, oscillate with the

same frequency

= (

−

)

/ ̄

and act as coupled dipole resonators. The phasing of their resulting

waves is such that energy is transferred from emitter to absorber at a rate that initially rises exponentially,

as shown in Fig. 6.

Energy transferred from one atom to another causes an increase in the minority state of the

superposition, thus increasing the dipole moment of both states and increasing the coupling and, hence, the

For the Transactional Interpretation[

], this phase selection process clarifies the mechanism by which, in the first stage of

transaction formation, the emitter makes a random choice between competing offer waves arriving from many potential

absorbers. The offer wave with the best phase wins, even if it come from far away. This process also clarifies the "hierarchy"

property of the TI: Among competing offer waves having the correct phase, those originating nearby will have larger initial

amplitudes and will be selected hierarchically over those that originate further away and are therefore weaker by 1/

19 of 34

rate of energy transfer. This self-reinforcing behavior gives the transition its initial exponential character,

and drives the transaction to its conclusion.

Thus, mutual offer/confirmation perturbations of emitter and absorber acting on each other create a

frequency-matched pair of dipole resonators as two-component superposed states, and this dynamically

unstable system either exponentially avalanches to the formation of a complete transaction, or it disappears

when a competing transaction forms elsewhere.

In a universe full of particles, this process does not occur in isolation, and both emitter and absorber

are also randomly perturbed by waves from other systems that can randomly drive the exponential

instability in either direction. This is the source of the intrinsic randomness in quantum processes. Ruth

Kastner[

] describes this intrinsic randomness as "spontaneous symmetry breaking", which perhaps

clarifies the process by analogy with quantum field theory.

We note here that the probability of the transition must depend on two things: the strength of the

electromagnetic coupling between the two states, and the degree to which the wave functions of the initial

states are superposed. The magnitude of the latter must depend on the environment, in which many other

atoms are "chattering" and inducing state-mixing perturbations. Thus, we see the emergence of

Fermi’s

"Golden Rule"[17]

, the assertion that a transition probability in a coupled quantum system depends on

the strength of the coupling and the density of states present to which the transition can proceed. Like the

emergence of the Born Rule from the Transactional Interpretation[

], the emergence of Fermi’s Golden

Rule is an unexpected gift delivered to us by the logic of the present formalism.

It is certainly not obvious

a priori

that the Schrödinger recipe for the vector potential in the momentum

(Eq. 22), together with the radiation law from a charge in motion (Eq. 33), would conspire to enable the

composition of the superposed states of two electromagnetically coupled wave functions to reinforce in

such a way that, from the asymmetrical starting state, the energy of one excited atom could

explosively and

completely

transfer to the unexcited atom, as shown in Fig. 6.

If nature had worked a slightly different way, an interaction between those atoms might have resulted

in a different phase, and no full transaction would have been possible. The fact that transfer of energy

between two atoms has this self-reinforcing character makes possible arrangements like a

laser

, where

many atoms in various states of excitation participate in a glorious dance, all participating at exactly the

same frequency and locked in phase.

Why do the signs come out that way?

No one has the slightest idea, but the behavior is so remarkable

that it has been given a name: Photons are classified as

bosons

, meaning that they behave that way!

That remarkable behavior is not due to any "particle" nature of the electromagnetic field, but to the

quantum nature of the states of electrons in atoms, and how the movement of an electron in a superposed

state couples to another such electron electromagnetically. The statistical QM formulation needed some

mechanism to finalize a transaction and did not recognize the inherent positive-feedback that nature built

into a pair of coupled wave functions. Therefore, the founders had to put in wave-function collapse "by

hand", and it has mystified the field ever since.

9. Two Atoms at a Distance

We saw that for two atoms to exchange energy, the field

at atom

must come from atom

, the

field

at atom

must come from atom

, and the phases of the oscillations must stay in coherent phase

with a particular phase relation during the entire transition.

This phase relation must be maintained even

which the two atoms are an arbitrary distance apart.

This is the problem we now address.

20 of 34

As before, we consider all electron motions and fields in the

direction, and drop the

subscript. To

be definite, we consider the case where the two atoms are separated along the

axis, atom

0 in the

excited state and atom

in its ground state, so their separation

is orthogonal to the

-directed

current in the atoms. The "light travel time" from atom

to atom

is thus

∆

. What is observed is

that the energy radiated by atom

at time

is absorbed by atom

at time

∆

∂

∣

∆

−

∂

∣

(50)

This behavior is familiar from the behavior of a "particle", which carries its own degrees of freedom with

it: It leaves

0 at time

and arrives at

at time

∆

after traveling at velocity

. Thus, Lewis’

"photon" became widely accepted as just another particle, with degrees of freedom of its own. We shall see

that this assumption violates a wide range of experimental findings.

For atom

, Eq. 43 becomes:

∂

∣

∆

∫

−

(

∆

)

∂

〈

〉

∂

∣

∆

(

∆

)

(51)

The retarded field from atom

interacts with the motion of the electron in atom

. The only difference

from our zero-delay solution is that the energy transfer has its time origin shifted by

∆

This result has required that we choose a

positive

sign for the

in Eq. 33. By doing that, we are building

in an "arrow of time", a preferred time direction, in the otherwise even-handed formulation. In particular

we are assuming that the retarded solution transfers positive energy. So far everything is familiar and

consistent with commonly held notions. However

the standard picture leaves no way for atom

to lose energy

to atom

- It does not conserve energy!

When energy is transferred between two atoms, the field amplitude must be "

coordinate and

symmetrical

" as Lewis described. The field

(

)

at the second atom due to the current in the

first must be exactly equal in magnitude to the the field

(

)

at the first atom due to the current in

the second, but separated in time by

∆

: For atom

, Eq. 43 becomes

∂

∣

∫

−

(

)

∂

〈

〉

∂

∣

(52)

So the field

from atom

, which arises from the motion of its electron at time

∆

, must arrive at

atom

at time

earlier

than its motion by

∆

. The only field that fulfills this condition is the

advanced

field from atom

, signified by choosing a negative sign for the

in Eq. 33. That choice uniquely

satisfies the requirement for conservation of energy. It also builds complementary "arrows of time" into the

formulation—we assume the advanced solution that transfers negative energy. These two assumptions

create a new "handshake" symmetry that is not expressed in conventional Maxwell E&M.

Once these choices for the

in Eq. 33 are made, the resulting equations for each of the energy

derivatives in Eq. 44 are unchanged when

∆

is substituted for

in the expression for

∂

. So each

transition proceeds in the local time frame of its atom—for all the world as if (except for amplitude) the

other atom were local to it. This "locality on the light cone" is the meaning of Lewis’ comment:

In a pure geometry it would surprise us to find that a true theorem becomes false when the page upon

which the figure is drawn is turned upside down. A dissymmetry alien to the pure geometry of relativity

has been introduced by our notion of causality.

21 of 34

The dissymmetry that concerned Lewis has been eliminated. This conclusion is completely consistent

with the 1909 formulation of Einstein, who was critical of the common practice of simply ignoring the

advanced solutions for electromagnetic propagation:

I regard the equations containing retarded functions, in contrast to Mr. Ritz, as merely auxiliary

mathematical forms. The reason I see myself compelled to take this view is first of all that those forms do

not subsume the energy principle, while I believe that we should adhere to the strict validity of the energy

principle until we have found important reasons for renouncing this guiding star.

After defining the retarded solution as

, and the advanced solution as

, he elaborates:

Setting

(

) =

amounts to calculating the electromagnetic effect at the point

from those

motions and configurations of the electric quantities that took place prior to the time point t

Setting

(

) =

one determines the electromagnetic effects from the motions and configurations

that take place after the time point t

In the first case the electric field is calculated from the totality of the processes producing it,

and in the second case from the totality of the processes absorbing it...

Both kinds of representation can always be used, regardless of how distant the absorbing bodies are

imagined to be. [22]

The choice of advanced or retarded solution cannot be made

a priori

: It depends upon the

boundary

conditions

of the particular problem at hand. The quantum exchange of energy between two atoms just

happens to require one advanced solution carrying negative energy and one retarded solution carrying

positive energy to satisfy its boundary conditions at the two atoms, which then guarantees the conservation

of energy.

Thus, the even-handed time symmetry of Wheeler-Feynman electrodynamics[

] and of the

Transactional Interpretation of quantum mechanics[

], as most simply personified in the two-atom photon

transaction considered here, arises from the symmetry of the electromagnetic propagation equations, with

boundary conditions imposed by the solution of the Schrödinger equation for the electron in each of the

two atoms, as foreseen by Schrödinger. We see that the missing ingredients in previous failed attempts, by

Schrödinger and others, to derive wave function collapse from the wave mechanics formalism were:

1. Advanced waves were not explicitly used as a part of the process.

The self-reinforcing nature of the interaction of two phase-locked wave functions coupled

electromagnetically, which drives the transition to completion, was not appreciated.

To keep in touch with experimental reality, we can estimate the "transition time"

from Eq. 46 and

Eq. 48:

∂

αβ

(53)

As we did for Eq. 39, from the green curve in Fig. 5 we can estimate the dipole matrix element, which is

times the "length" between the positive and negative "charge lumps", say

≈

. At the steepest part

of the transition, all the

and

terms will be 1/

√

2, so

∂

∣

max

≈

(

)

(54)

From any treatment of the hydrogen spectrum we obtain, for the 210

→

100 transition:

128

πe

(55)