rmead@whale.st.usm.ed

The transactional interpretation of quantum mechanics (J.G.
Cramer, *Phys. Rev. D 22*, 362 (1980) ) has received little
attention over the one and one half decades since its conception.
It is to be emphasized that, like the Many-Worlds and other
interpretations, the transactional interpretation (TI) makes no new
physical predictions; it merely reinterprets the physical content
of the very same mathematical formalism as used in the ``standard''
textbooks, or by all other interpretations.

The following summarizes the TI. Consider a two-body system (there are no additional complications arising in the many-body case); the quantum mechanical object located at space-time point (R_1,T_1) and another with which it will interact at (R_2,T_2). A quantum mechanical process governed by E=h\nu, conservation laws, etc., occurs between the two in the following way.

1) The ``emitter'' (E) at (R_1,T_1) emits a retarded ``offer wave'' (OW) \Psi. This wave (or state vector) is an actual physical wave and not (as in the Copenhagen interpretation) just a ``probability'' wave.

2) The ``absorber'' (A) at (R_2,T_2) receives the OW and is
stimulated to emit an **advanced** ``echo'' or ``confirmation
wave'' (CW) proportional to \Psi at R_2 backward in time; the
proportionality factor is \Psi* (R_2,T_2).

3) The advanced wave which arrives at 'E' is \Psi \Psi* and is presumed to be the probability, P, that the transaction is complete (ie., that an interaction has taken place).

4) The exchange of OW's and CW's continues until a net exchange
of energy and other conserved quantities occurs dictated by the
quantum boundary conditions of the system, at which point the
``transaction'' is complete. In effect, a standing wave in
**space-time** is set up between 'E' and 'A', consistent with
conservation of energy and momentum (and angular momentum). The
formation of this superposition of advanced and retarded waves is
the equivalent to the Copenhagen ``collapse of the state vector''.
An observer perceives only the completed transaction, however,
which he would interpret as a single, retarded wave (photon, for
example) traveling from 'E' to 'A'.

Q1. When does the ``collapse'' occur?

A1. This is no longer a meaningful question. The quantum
measurement process happens ``when'' the transaction (OW sent - CW
received - standing wave formed with probability \Psi \Psi*) is
finished - and this happens over a **space-time** interval;
thus, one cannot point to a time of collapse, only to an interval
of collapse (consistent with relativity).

Q2. Wait a moment. What you are describing is time reversal invariant. But for a massive particle you have to use the Schrodinger equation and if \Psi is a solution (OW), then \Psi* is not a solution. What gives?

A2. Remember that the CW must be time-reversed, and in general must be relativistically invariant; ie., a solution of the Dirac equation. Now (eg., see Bjorken and Drell, Relativistic QM), the nonrelativistic limit of that is not just the Schrodinger equation, but two Schrodinger equations: the time forward equation satisfied by \Psi, and the time reversed Schrodinger equation (which has i --> -i) for which \Psi* is the correct solution. Thus, \Psi* is the correct CW for \Psi as the OW.

Q3. What about other objects in other places?

A3. The whole process is three dimensional (space). The retarded OW is sent in all spatial directions. Other objects receiving the OW are sending back their own CW advanced waves to 'E' also. Suppose the receivers are labeled 1 and 2, with corresponding energy changes E_1 and E_2. Then the state vector of the system could be written as a superposition of waves in the standard fashion. In particular, two possible transactions could form: exchange of energy E_1 with probability P_1=\Psi_1 \Psi_1*, or E_2 with probability P_2=\Psi_2 \Psi_2*. Here, the conjugated waves are the advanced waves evaluated at the position of R_1 or R_2 respectively according to rule 3 above.

Q4. Involving as it does an entire space-time interval, isn't this a nonlocal ``theory''?

A4. Yes, indeed; it was explicitly designed that way. As you
know from Bell's theorem, no ``theory'' can agree with quantum
mechanics unless it is nonlocal in character. In effect, the TI is
a hidden variables theory as it postulates a **real** waves
traveling in space-time.

Q5. What happens to OW's that are not ``absorbed'' ?

A5. Inasmuch as they do not stimulate a responsive CW, they just continue to travel onward until they do. This does not present any problems since in that case no energy or momentum or any other physical observable is transferred.

Q6. How about all of the standard measurement thought experiments like the EPR, Schrodinger's cat, Wigner's friend, and Renninger's negative-result experiment?

A6. The interpretational difficulties with the latter three are due to the necessity of deciding when the Copenhagen state reduction occurs. As we saw above, in the TI there is no specific time when the transaction is complete. The EPR is a completeness argument requiring objective reality. The TI supplies this as well; the OW and CW are real waves, not waves of probability.

Q7. I am curious about more technical details. Can you give a further reference?

A7. If you understand the theory of ``advanced'' and
``retarded'' waves (out of electromagnetism and optics), many of
the details of TI calculations can be found in: *Reviews of
Modern Physics*, Vol. 58, July 1986, pp. 647-687 available on
the WWW as: http://mist.npl.washington.edu/npl/int_rep/tiqm/TI_toc.html

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