On the Relativity of Quantum Superpositions
            ___________________________________________
 
                           Paul Merriam 
                          Santa Cruz, CA 
                       pmerriam@cruzio.com 

                    (received: June 23, 1997)


Quantum Mechanics poses serious problems for ontologists.  Among these 
problems is the fact that a quantum system does not seem to have a sensible 
state of existence *until* it is observed.  Thus it is not clear how to 
reconcile quantum mechanics with usual scientific metaphysics.  In this paper 
it is suggested that the state of a quantum system is partially a function of 
which system is doing the describing, analogous to differing frames of 
reference in relativity. It is argued that this `relative superposition' 
notion is a necessary stance in quantum philosophy. 
                  ____________________________________
 

1.  Consideration of a coupled system
_____________________________________
 
	To show how quantum mechanics is different from other scientific 
theories we will focus on a bizarre aspect of the "measurement problem", which 
is often crucial in interpretations of quantum mechanics.  In this discussion 
I observe Jaunch [1968] and Hughes [1989] (ch. 6, sec. 9 and ch. 9 sec. 6, 
respectively).
 
	In the following considerations I want to talk about the state of a
quantum mechanically described cat (and experimenter) 
for the sake of conceptual 
ease. In practice macroscopic normal-temperature object will not remain in a
superposition state for very long, but neither Schroedinger's equation (or its
relativistic generalizations) nor Luder's rule (or von Neumann's projection
postulate) refer to the *size*, *complexity*, or *consciousness* of the systems
they describe.  As a result, I conclude there is no difference *as a matter of
principle* between the behavior of a quantum-mechanical "cat" and a
quantum-mechanical "electron".  In this paper we will be concerned with 
questions of principle, as opposed to the practicality of this or that 
experimental situation.

	The situation is as follows:  Schroedinger's cat C is in the usual 
(idealized) quantum-mechanically sealed-off box. There is an experimenter (or 
experimental device) called E1 who is inside a laboratory and conducting the 
experiment on the cat.  We are outside the lab, which is itself perfectly 
quantum-mechanically sealed off (from us.)  What does quantum mechanics tell 
us about the experiment E1 is conducting? 

	Assume the observable being measured by E1 is representable by the 
operator `C' on the Hilbert space `H(C)' of Schroedinger's cat.  Assume there 
are just two values of the observable `C', eignvalues of the eignvectors `v+' 
and `v-'.  The experimenter (or measuring device) `E1' is assumed to have (at 
least) three mutually exclusive possible states, an initial state uo before 
any measurement takes place, `u1' a state when the experimenter registers a 
positive value for `C', and `u2' a state when it registers a negative value 
for `C'.  We assume the quantum mechanical formalism can be applied to the 
experimenter (or device), and that these three states are representable by 
vectors `uo', `u+', and `u-' in a Hilbert space `H(E1)'. No assumption is made 
*a priori* that superpositions of `uo', `u+', and `u-' are also possible states 
of the experimenter's system `E1'.
 
	The state of the coupled system C+E1 is represented in the 
tensor-product space H(C) x H(E1).  Assume before the measurement begins the 
system `C' is in the pure state represented by the vector    v = c+v+ + c-v-   
and that E1 is in the state (represented by) `uo'. Then the original state 
of C+E1 will be Psi(0) = v x u0 .  During the course of the measurement 
interaction this state will evolve continuously according to Schroedinger's 
equation.  Accordingly, at the end of the interaction, C+E1 will again be in a 
pure state Psi = U Psi(0) which is an element of H(c) x H(E1)  with `U' some 
unitary operator on H(c) x H(E1).  `U' must obey the following constraints:  
when v = v+ (that is, when c- = 0), we require that Psi = Psi+ = v+ x u+; when 
v = v- (that is, when c+ = 0), we require that Psi = Psi- = v- x u-.
 
(editor's note: x is the vector cross-product)
 
	In each of these two cases `U' takes Psi(0) into a state of C+E1 
reducible into a pure state of `C' and the corresponding pure state of `E'.  
By the linearity of `U' we obtain
 
                         Psi = c+ Psi+ + c- Psi-
 
for the general case.  However, although this is a pure state of C+E1, this 
state is in general not reducible to pure states of C and of E1.
 
	If the experimenter *actually* has found the cat to be either alive or 
dead, then `C' is either in state v+ or in state v-, and E1 is therefore in 
the correlated state u+ or u-.  The state of the composite system is then 
either v+ x u+ or v- x u-,  each of these having a certain probability.
 
	But this means that C+E1 is in a mixture, contradicting the fact that 
according to quantum mechanics C+E1 is in a pure state: the superposition Psi. 
Note that we have not "collapsed the statevector" of the laboratory by (for 
example) opening the door separating us from E1.  It is possible to get more 
specific here, but the philosophical point has been made.  In sum:
 
(1.1)	According to someone outside the lab system, the state of  the 
lab is Psi.
 
(1.2)	According to someone inside the lab system, the lab is either in 
the state v+ x u+  or in the state v- x u-.
 
(1.3)	The states in (1.1) and (1.2) are not the same states. 	
 

2.  Philosophical conclusions
_____________________________
 
*What does this mean?*
 
	(1.3) is a problem at the heart of quantum theory and cannot be "swept 
under the rug". One might try to claim that a "Heisenberg cut" (a dividing 
line between micro- and macro-scopic scales) could resolve the issue, but then 
one is committed to explaining microscopic phenomena via macroscopic objects 
(such as the experimental apparatus,) which is untenable as a fundamental 
description of the world.
 
	To make this situation even more immediate we may consider things from 
the perspective of the experimenter E1.  To E1 the cat as well as we who are 
outside the laboratory are quantum mechanically sealed off--and as elements of 
non-interacting systems evolve via unitary evolution.  If there is more than 
one possible state that E1 may observe us to be in, than according to the 
quantum predictions (*and experiments*) of E1 we evolve into a superposition 
of states.
 
	Label any experimenters (or experimental equipment) outside of the 
laboratory as E2. Then for E2, E1 evolves into a superposition of states--and 
for E1,  E2 evolves into a superposition of states.  It must be taken 
seriously:  there is no reason as far as quantum mechanics *itself* is 
concerned why we should not actually consider the case of experimenters in 
this situation.  "But, if I am any judge, I am not in a superposition 
state"--true, but quantum mechanics does *not* tell us that a system could 
perceive *itself* to be in a superposition state, only that *unobserved* 
systems may be.  In light of section 1, what will we find if we stick strictly 
with the quantum mechanical formalism?
 
	Referring to section 1, we may say _without philosophical bias_ that 
*experimenter 2's* account of E1's experiment and *experimenter 1's* own 
account are different:  the experimenter, no different than anyone else, 
perceives him/herself as definitely being in a pure state at the end of the 
experiment (either u1 or u2, represented by u+ or u- respectively)--while we 
just proved that for someone standing outside the lab (E2) this is not true: 
the experimenter remains entangled with the cat.  For E2 it is 
*experimentally* true that E1 remains entangled with the cat.  In deriving 
this conclusion we have not overstepped the bounds of computational quantum 
mechanics in the least--we have merely taken it at face value as a 
calculational tool.
 
	This situation is not as artificial or remote from experiment as might 
seem;  we are faced with the issue of an experiment let run half the night, 
whose results are then automatically printed onto paper.  Were the results 
actually "collapsed" onto the paper the next morning when the experimenter 
came into the lab to read them? 	 	

        Now consider the fact that the only assumption we made about the cat 
was that is was sealed off from the rest of the laboratory, and could be 
observed in one of two possible states (represented by v+ and v-).  Once 
again, what if E1 calculates E2 to be in a superposition?  Suppose E1 observes 
E2 to be in a superposition state Psi2. Notice (as one may easily check) that 
experimenter 2's formal treatment of the cat C in section 1 can be applied to 
*experimenter 2 itself* because E2  would be just giving quantum mechanical 
description of what E1 is giving a quantum description of in both cases.  
Since we end up with experimenter 1 in some superposition state Psi(1) we will 
now (by the arguments above) find experimenter 2 in the state Psi' = Psi(1) x 
Psi(2).  Yet when *I* am experimenter 2 I certainly do not observe myself to 
be in a superposition state!!
 
	What is going on?  What we have just demonstrated a case for which
 
(2.1) *_my_ calculation of the _experimenter's_ calculation of _me_ is some 
superposition state Psi', *
 
(2.2) *at the same time, _I_ observe _myself_ to be in v+ (or v-)--not a 
superposition state, *
 
(2.3) *the states in (2.1) and (2.2) are different states. *
 
If we include an initial state vo for E1's representation of my system, then 
the relation between my system (E2) and the experimenter's (E1) on the one hand, 
and the relation between the experimenter's and my own system on the other hand, 
are the *same*.  Call this relation `R'.  Using the argument of section 1 we will 
again, as one may check, arrive at (1.3).  Then formally we have
 
(2.4)  E2 R E1  and  E1 R E2   =/=>   E2 R E2
 
This is intransitivity.  The inevitable conclusion is therefore:
 
                             *Quantum theory is an intransitive theory.*
 
 
3.  Philosophical implications
______________________________
 
	What does it mean for a theory to be intransitive?  The answer is that 
the theorizer is not arbitrary:  what prediction is made depends on which 
system is doing the predicting. I cannot apply *my* quantum theory to *your* 
(distinct) system and calculate what reality is like *for you*.  If we believe 
our theories as descriptions of reality (and if we don't I don't see what the 
point of theorizing is) then we are compelled to the conclusion that the 
reality of the behavior of a physical object is well-defined only when 
relativised to a particular system.	
 
	If we believe quantum mechanics, then the reality of the cat *for the 
cat* is different then the reality of the cat *for the experimenter*.  There 
is no contradiction here so long as we concede that "objective reality" is (at 
least partially) a function of *quantum systems*. A quantum description of the 
state of a system is relative to the system doing the describing just as the 
description of a system in terms of space and time is relative to the motion 
(and gravity) of the describing system.  Hence it does not make sense to ask 
"what is the prediction of quantum mechanics?" since there is no single 
prediction--"the" prediction depends (crucially) on *which* system is doing 
the predicting.  It is more accurate to speak of an ontologically distinct 
(but identical) quantum mechanics for *each* system.
 
	An intransitive theory relates the world to *my* system, not the world 
to everything else in the world.  But if the interaction of a system with 
things outside the system are (at least partially) defined by the system 
itself then what we have is an inherently subjective view of physical 
interactions.  That we could have such a situation in objective science is 
surprisingly reasonable from the perspective of Occam's Razor: every system in 
an "objective" universe should, if science is going to work, be able to derive 
the same scientific laws.  These laws are, for each system, a theoretical 
construct based on (formally speaking) observations by *that* system.  The 
"hypotheses of parsimonious objective science" one might call it, is that each 
system may similarly correlate observations made by the systems *themselves*.  
It seems that theoretical transitivity is in fact an additional assumption 
(that we do not need to do objective science).
 
 
4.  Perspectivist quantum theory
________________________________
 
	A moment of reflection is in order.  In QM, we may have an 
*unobserved* electron, say, and we know the evolution of its state function 
obeys Schroedinger's equation.  In fact, the state function of *everything* 
unobserved behaves according to the Schroedinger equation, even though in 
practice it becomes impractical to solve for complicated systems. [1]
 
	Now there is already a subtlety here.  What does "we may have an 
unobserved electron" mean?  What it is taken to mean is that *we*, the 
macroscopic-conscious- complicated physicists-and-laboratories are not in a 
superposition (our whole system is perhaps in an eigenstate of the hugely 
complicated Hilbert space representing us) while the electron is (possibly) in 
a superposition--its wave function is not "collapsed" until it interacts with 
our system.  It is *because* the electron is not a part of our system that it 
can be in a superposition.
 
	And yet *nothing* in orthodox quantum mechanics refers to size, 
complexity, or consciousness.  This means, if we take the formalism of quantum 
mechanics at face value, there is no "ontological" difference between an 
experimenter and an electron so far as superpositions are concerned.  There is 
nothing *in principle* to prevent one from applying quantum mechanics *to the 
experimenter from the point of view of the electron* in the same way we have 
just applied QM *to* the electron *from* the point of view of the 
experimenter. What I mean is instead of regarding the experimenter/lab as 
"collapsed" (in a definite state) and the (non-interacting) electron as being 
in a superposition state, *there is nothing in the formalism of quantum 
mechanics itself* to prevent us from regarding the electron as being in a 
definite state and the experimenter/lab. as being in a superposition state.
 
*Objection*:  If the electron is in a definite state then no interference 
pattern will be produced.
 
*Reply*:  False, because the double-slit-plate-and-screen system are now in a 
(corresponding) superposition state (i.e. *relative* to the electron!)  Using 
the electron as reference-system, the *laboratory* is in a superposition.  
Normally we would say its wavelength is given by de Broglie lambda = h/p which 
is ~= 10^{-35}m, negligibly small.  Thus the de Broglie relation must be 
relativised to the mass of the system doing the describing. The laboratory is 
now in a superposition, Psi(lab) = Psi(1) x Psi(2), as a result of which it 
may take on any state | Psi(lab) > = c1 | Psi(1) > + c2 | Psi(2) >. Whether we 
think of the superposed electron evolving within the laboratory or the 
superposed lab moving past the electron does not make any difference as far as 
experimental predictions are concerned, so long as the wave equations 
describing the laboratory from the electron's point of view are scaled 
appropriately. 	 

More generally: 	
	If the situation is described by system A is that system B is in 
state Psi, given in terms of basis vectors of the appropriate Hilbert space
 
		Psi = c1 v1 + c2 v2 + ...
 
with the probability of observation of B being in eigenstate vi given by 
|c(i)|^2 then (for the same situation) B describes A as
 
		Psi' = c'1 v'1 + c'2 v'2 + ...
 
where the following hold:
 
	1.  If A observes B in state vi, then B observes A in state v'i from 
which we see that the probabilities must be the same because of the (presumed) 
possibility of observation at any moment:
 
	2. |c'i|^2 = |ci|^2
 
Because of phase-sensitive experiments (like the Aharonov-Bohm-effect etc.) 
we in fact have
 
	3.  c'i = +/- ci
 
So far as temporal evolution is concerned, we have
 
		Psi(t) = U(t) Psi(0)
 
and 		Psi'(t) = U'(t) Psi'(0)
 
with U = U' because the probabilities must co-evolve.
 
	In fact, it is of some interest to note that (for humans) all 
experimental results ultimately come back to a human observer (via observing 
an automated printout or whatever),  which means that regarding ourselves as 
being in a *definite* eigenstate *ontologically* (i.e. relative to every 
system) is an approximation valid only to our de Broglie wavelength--which is 
very close to the Plank-length.
 

5.  On the relativity of quantum systems
________________________________________
 
	To emphasize the naturalness of this stance on quantum philosophy as well 
as the analogy to special relativity I here reproduce the beginning of the
original special relativity paper of Albert Einstein, merely replacing phrases
like "relative velocity" with "relative quantum state" and "relativistic
observation" with "quantum observation."  The almost seamless ability to do this 
is striking.  There is no new information in this section however, so the reader 
may wish to skip it.     
 
	We know from section 1 that quantum mechanics--when applied to 
different systems, leads to asymmetries which do not appear to be inherent in 
the phenomena.  Take, for example, the reciprocal state of superpositions of a 
lab Psi(lab) and outside experimenter Psi(exp).  The observable phenomenon 
here depends only on the relative state of the lab and experimenter, whereas 
the customary view draws a sharp distinction between the two cases in which 
either the one or the other of these systems is *unobserved*.  For if the 
laboratory is unobserved in the quantum-mechanical sense and the experimenter 
has a definite "collapsed" state, there arises a superposition of possible 
states of observation of the laboratory.  But if the laboratory is in a 
definite "collapsed" state and the *experimenter* is unobserved, no 
superposition of possible states arises for the laboratory. For the 
experimenter, however, we find a superposition of states, to which in itself 
there is not necessarily corresponding states, but which gives rise--assuming 
the same ontological situation in the two cases discussed--to the same 
observed states of interaction as those in the former case.

	Examples of this sort, together with the unsuccessful attempts to 
discover any criteria for a "Heisenberg cut" relative to the "space-time 
continuum," suggest that the phenomena of quantum-mechanics as well as of 
relativity possess no properties corresponding to the idea of absolute state.  
They suggest rather that, as has already been shown to the first order of 
small systems, the same laws of quantum mechanics and relativity will be valid 
for all observing systems for which the equations of physics hold good.  We 
will raise this conjecture (the purport of which will hereafter be called the 
"Principle of Perspective") to the status of a postulate, and also introduce 
another postulate, which is only apparently irreconcilable with the former, 
namely, that "objective" physical properties are actually the subjective 
states of interaction of physical systems, which accords with some theories of 
mind.  These two postulates suffice for the attainment of a simple and 
consistent theory of the quantum dynamics of superposition states and 
relativity based on the subjective "interaction" states of different systems.  
The introduction of a "space-time continuum" will prove to be superfluous 
inasmuch as the view here to be developed will not require a "background 
manifold" provided with special properties, nor assign a superposition state 
to curvatures of any kind.  The theory to be developed is ultimately 
based--like all conceptual physics--on the interactions between definite 
objects, since the assertions of any such theory have to do with the 
relationships between various systems.  Insufficient consideration of this 
circumstance lies at the root of the difficulties which the unified theories 
of physics at present encounter.
 
 
                   ?I.  Relativity of Relativity
 
	Let us take a "collapsed" system from which the equations of quantum 
mechanics hold good [2].   In order to render our presentation more precise 
and to distinguish this system of observation verbally from others which will 
be introduced hereafter, we call it the "observing system."
 
	If a material point is *in the process* of being observed relative to 
this system of observation, its state can be defined relative thereto by the 
employment of the methods of relativity and can be expressed in Minkowskian 
coordinates.
 
        If we wish to describe the *motion* of a material point, we give the 
potential values of its coordinates as functions of time.  Now we must bear 
carefully in mind that a mathematical description of this kind has no physical 
meaning unless we are quite clear as to what we understand by "space-time 
manifold"  We have to take into account that all our judgments in which 
space-time plays a part are always judgments of simultaneously 
quantum-mechanically observed events.  If, for instance, I say, "That train 
arrives here at 7 o'clock," I mean something like this: "The pointing of the 
small hand of my watch to 7 and the arrival of the train are simultaneously 
*quantum-mechanically observed* by *me*."
 
	It might appear possible to overcome all the difficulties attending 
the definition of "space-time event" (with respect to the co-consistency of 
quantum mechanics) by substituting "the quantum-mechanical observation of the 
small hand of my watch" for "space-time event."  And in fact such a definition 
is satisfactory when we are concerned with defining a space-time continuum 
exclusively for the quantum system from which the watch is within; but it is 
no longer satisfactory when we have to connect in quantum mechanics space-time 
continuums occurring for different systems, or--what comes to the same 
thing--to evaluate the set of events occurring for systems 
quantum-mechanically uncorrelated with the watch.
 
	We might, of course, content ourselves with space-times determined by 
an observer in some one quantum-mechanical system using that system as the 
*definer* of the quantum- mechanical universe, and coordinating the 
corresponding quantum-mechanical observations of the watch-hands with 
observations, realized by every space-time event to be accounted for, and 
reaching them through uninterrupted observation.  But this co- ordination has 
the disadvantage that it is not independent of the standpoint of the observer 
(system) of the watch or clock, as we know from (1.3).  We arrive at a much 
more practical determination along the following line of thought.
 
	If there is some quantum mechanical system A, the observer A can 
determine the physics of things relating to A itself by making quantum 
mechanical (qm) observations. If there is another (quantum mechanically 
unconnected) system B in all respects resembling A, it is possible for an 
observer at B to determine the experimental correlations of qm events within 
the set of qm observations of B.  But it is not possible without further 
assumption to compare, in respect of superpositions, an observation of A with 
an observation of B.  We have so far defined only a set of "A correlations" 
and set of "B correlations," we have not defined a common "quantum 
mechanically predicted state of a system" or even "probability density" for A 
and B, for the latter cannot be defined at all unless we establish *by 
definition* that the "state of a system" required by observations from A 
corresponds to the "interactions" observed by B.  Let, A having just observed 
B to be in the eigenstate |b(0)>, B's state vector evolve according to 
Schroedinger:
 
		(i hbar) (d Psi)/(d t) = H Psi
 
in which the "t" is the classical time t(A) of system A.  After Delta-t(A) 
system A again observes system B, which is now observed to be in state |b(1)>.  
We can now give an identical treatment from the perspective of B this time 
using t(B).  In accordance with definition the two systems quantum mechanically 
co-relate (i.e. may belong to a single description) if
 
                 O'(A) - O(A) = O'(B) - O(B)
 
every interval as measured by A and B coincides...	
 
 
6.  Conclusion
______________
 
	We have shown that quantum mechanics is an intransitive theory, but 
this does not necessitate mystical solutions to quantum problems.  A 
relativity of quantum states is possible, in analogy to distance and time 
values in special relativity.  This may lead us to a natural ontological 
philosophy of quantum mechanics.
 
                   ______________________________
 
Footnote:
_________
 
[1]   I am taking some liberty with the terminology here.  For this informal 
part of the discussion one may simply take a system's evolving according to 
Schroedinger's equation as the definition of it being "unobserved".
 
[2]   This is normally the laboratory system.
 

References:
___________
 
Einstein, A. [1905]: 'On the Electrodynamics of Moving Bodies.' The Principle 
of Relativity
 
Hughes, R. I. G. [1989] The Structure and Interpretation of Quantum Mechanics 
(Harvard University Press, Cambridge, M.A.)
 
Jaunch, J. M. [1968] Foundations of Quantum Mechanics (Addison Wesley, 
Reading, MA)
 
von Neumann, J. [1955] Mathematical Foundations of Quantum Mechanics 
(Princeton University Press, Princeton, N.J.)