(Daniel Weston) wrote in message ...
Patrick, you keep saying that QM is a principle theory, not a
constructive theory.
Could you explain this conclusion. My interpretation of Einstein's
comments that you have given us, is quite different. Thanks.

************************************************** ************

Is QM a principle theory?

In the first place, all classifications are arbitrary. In the second
place, I know that some posters feel that the "principled vs
constructive" distinction serves no useful purpose in physics. I
disagree and I'll make clear exactly why I disagree later. The central
point of contention in the distinction between principle or
constructive theories is the existence of a "hypothetical" ontological
element in the theory. What is "hypothetical" and what is not as a
physical concept is time dependent and established by convention
(Poincare) of the physics community. Einstein's original idea of a
hypothetical ontological element was this: If it's assumed to exist,
but invisible, it's hypothetical. However, as useful as this notion
was a 100 years ago, it's too impractical now. For practicality,
instead of saying the "hypothetical electron" we just say the
"electron," and hopefully remember that electrons are hypothetical in
some ontological sense.

I have chosen the term "ontological element" carefully. By "ontology"
I mean existence in a material sense. To Einstein, the concepts of the
luminiferous ether and the atom, if they should correspond to anything
"real" at all, correspond to things materially "real," that is, they
correspond to something made of matter. Ordinary matter, whatever it
is metaphysically, is well describable physically: It has mass (get
enough of it and you can weigh it); it has inertial properties; it is
affected by the gravitational fields of other matter particles. Maybe
you can think of more. OK, enough introduction.

Let's begin with a quote from Einstein:

We can distinguish various kinds of theories
in physics. Most of them are constructive.
They attempt to build up a picture of the more
complex phenomena out of the materials of a
relatively simple formal scheme from which
they start out. Thus the kinetic theory of gases
seeks to reduce mechanical, thermal, and
diffusional processes to movements of molecules
-- i.e., to build them up out of the hypothesis of
molecular motion. When we say that we have
succeeded in understanding a group of natural
processes we invariably mean that a constructive
theory has been found which covers the
processes in question.
Along with this most important class of
theories there exists a second, which I will
call 'principle-theories'; These employ the
analytic, not the synthetic, method. The elements
which form their bases and starting-point are not
hypothetically constructed but empirically
discovered ones, general characteristics of
natural processes, principles that give rise to
mathematically formulated criteria which these
separate processes or the theoretical
representations of them have to satisfy. Thus
the science of thermodynamics seeks by
analytical means to deduce necessary conditions,
which separate events have to satisfy, from the
universally experienced fact that perpetual
motion is impossible.
The advantages of the constructive theory
are completeness, adaptability, and clearness,
those of the principle theory are logical
perfection and security of the foundations.
The theory of relativity belongs to the latter
class. In order to grasp its nature, one needs
first of all to become acquainted with the
principles on which it is based. Before I go
into these, however, I must observe that the
theory of relativity resembles a building \0
consisting of two separate stories, the special
theory and the general theory. The special
theory, on which the general theory rests,
applies to all physical phenomena with the
exception of gravitation; the general theory
provides the law of gravitation and its relations
to the other forces of nature.
Found in: "What is the Theory of Relativity?",
Einstein, Ideas and Opinions, Three Rivers
Press, p. 228-9.

I have written much on this quote on my website. See

http://www.ajnpx.com/html/Science/Ho...long-haul.html

http://www.ajnpx.com/html/Einstein's-search-for-meaningful-unification-in-physics.html


I can at this point only do two things to clear up what Einstein meant
by "principle vs constructive theories": First, I will place his
characterizations of them in his time period for a context, providing
his insight into Newton's mechanics as a starting point. Second, I
will attempt to clarify what I think can be made into a doable and
useful definition of these two terms. But first a definition. A
"principle" is a law or heuristic in which one has great confidence.

Einstein told us that he was once an etherist. He (apparently)
accepted the Mechanical Program's dogma to explain all phenomena in
terms of mechanics? But what is mechanics? Mechanics is the science
that MODELS all matter as being aggregates of point mass particles in
motion. Newton's mechanics adds to that description that masses
interact via forces acting-at-a-distance. (Hertz's mechanics --
inspired by his positivist leanings -- does not use the force concept
as a primitive.) The logical extension to this rigid approach would be
to approximate matter as a continuum state. The mathematical
distinction between these two approaches was profound, for in the
former case one uses total differential equations and in the latter
one uses partial differential equations. Einstein told us that this
led the way for the development of field theories of electrodynamics.

But a new way to conceive of physics arose as a result of the
investigations of Faraday, Maxwell, and Lorentz, which was the field
concept of interactions over space -- with changes to fields
propagating at finite speeds. Einstein also told us that the physicist
at the end of the 19th century had come to treat the EM fields as
purely irreducible, meaning that they ceased to attach any physical
meaning about them as "being states of some material ether" ---
unless, of course, you were Einstein trying to lay a foundation for
all of physics. This effort was to lead him to despair of the entire
mechanical, constructivist approach to foundations in physics and to
lead a whole generation of physicists into a new way to think about
theory invention for the future. Because theory invention is
important, the distinction between principle vs constructive theories
is important.

I'm going to be very speculative myself here about Einstein's motives.
I have defined physics as the search for the smallest set of rules
that completely describes the behavior of the inanimate material realm
under natural conditions. My speculating is that Einstein would have
liked this definition. I am going to proceed under this assumption. If
I don't do this, there is no way I can find a satisfactory explanation
of Einstein's use of the term "hypothetical" to his characterizations.
The material world "exists" in two primary forms: that which can be
seen by the naked eye, and that which cannot. That which can be seen
by the naked eye is NOT hypothetical! (From the ontological standpoint
of Einstein's metaphysics, this is the primary bootstrapping step.)
However, we cannot see atoms or molecules or their paths through space
with the unaided eye. The very existence and motions of molecules and
theory motions through space are thus "hypothetical." Here's where the
distinction making gets deep: Sure, molecules are "hypothetical," but
they're "hypothetical" real in the sense that they are conceivable as
limiting cases of visible matter simply made invisibly small. On the
other hand, field doesn't even have that status. Field is not a
limiting case of something visible. Field is purely speculative by
these standards, a free creation of the human mind.

To Einstein of 1905 prior to SR, the universe was modeled as a
collection of three things: matter, fields by which matter interacts
with matter, and absolute space (one leftover from Newton's mechanics
and one possible space -- the rest frame of the luminiferous ether).
Einstein would call SR a principle theory, yet it is an extension to
Newtonian mechanics. So what's the status of Newtonian mechanics? Is
it principled or constructive. Well, obviously it is constructive. It
is founded on the assumption that matter is modeled as aggregates of
this hypothetical thing called the point mass particle. The kinetic
theory (the original statistical mechanics) is also a constructive
theory for much the same reason. But classical thermodynamics --
Einstein's inspiration for how to design SR -- is a principle theory.
It's not based on assumptions of what matter is made out of in the
small range of existence, but, rather, is a theory built directly on
equations (principles) that account for thermodynamic measurements.

So, is SR really a principle theory if it is a generalization of the
constructive theory of Newtonian mechanics. I say, no! As I have
claimed many times, I do not always agree with Einstein. But, I can go
so far as to claim that SR is an principled "extension" to Newtonian
mechanics. For SR still retains that hypothetical point mass particle
of Newton's mechanics, though it adds into the mix the principle of
electrical charge and with it Maxwell's equations, the PoR, and the
Light Principle. In its weak form the PoR is this: the general laws of
physics (assuming such a concept of "general laws" is meaningful at
all) have the same form in all inertial frames of reference.

There is one more principle that people tend to overlook that is in
that extension from Newtonian mechanics to SR: Which is the "Principle
of Generalization" (PoG), which I coined. What if the unit of measure
could randomly change in time in a given inertial reference frame?
Such a weird occurrence could not be noticeable within that frame
though. It's conceivable that the laws of physics could have the same
form in all inertial frames of reference, yet there be no way to
relate the laws between frames as is done with the Galilean or Lorentz
transformations. Which would mean that the PoR could not be of any use
as a heuristic for the discovering of new general laws of physics.
However, our experience of comparing measurements made among different
frames of reference does not lead us to believe that the units of
measure of different frames are randomly changing to any noticeable
degree. So, we have the justification for believing not only that
there a laws of the same form in all inertial frames, but also that
they are relatable somehow under spacetime transformations of
coordinates. So, the PoG would be that any generalization of Newtonian
mechanics must 1) not add to the number of absolute spaces, and 2)
must have general laws which satisfy a covariance in the extended
theory, and which reduces to Galilean covariance in the limit that the
theory is in the "Newtonian domain of applicability."

OK, we've now dealt with the meaning of "hypothetical" in Einstein's
passage above. We have seen some examples of constructive and
principled theories. But there's a bit more distinction making to do:

Along with this most important class of
theories there exists a second, which I will
call 'principle-theories'; These employ the
analytic, not the synthetic, method. The elements
which form their bases and starting-point are not
hypothetically constructed but empirically
discovered ones, general characteristics of
natural processes, principles that give rise to
mathematically formulated criteria which these
separate processes or the theoretical
representations of them have to satisfy.

What Einstein means here is simple enough for me, after I have now
studied it for a long time. The point of either kind of theory --
constructive of principled -- is to say something about the visible
material realm, because that's the testable realm in which the naked
eye and all other measuring instruments exist. The distinction between
these two kinds of theories, according to Einstein, is on how one
forms the starting point! If you start with empirically induced
principles about heuristics, laws, and/or measurements themselves and
have avoided the ontological hypotheticalness of the invisible realm,
you have created a principled theory. On the other hand, if you want
to start by explaining the properties of the macroscopic realm in
terms of hypothetical invisible ontological elements, you have created
a constructive theory, by Einstein's characterizations.

Einstein claimed that the PoR and the Light Principle are empirically
deduced (I would say "induced"), not hypothetically supposed. I have
no problem with that. But I don't like his distinctions between
"analytic" and "synthetic." He could have left these terms out
altogether and lost nothing in the characterization. This is all I can
say about them. It appears that Einstein means by "analytic" that one
starts with empirically induced laws or heuristics and makes logical
deductions on them to arrive at empirically testable conclusions. On
the other hand, "synthesis" means "putting together." So, I suppose
Einstein meant by the use of the term "synthetic" that one builds up
the visible world out of hypothetical real stuff that is invisible.

So, finally, on to QM. I'm going to just use the Schroedinger theory
for my explanation. A lot happened between 1905 and 1926. The
existence of the electron and proton were no longer considered
"hypothetical" by 1926. Physics had evolved to accept the existence of
the electron and proton as a convention. Thus we can legitimately talk
about principle theories of the hydrogen atom. Now, by the standards
of the conventions held in 1926 there were in Schroedinger's theory no
hypothetical ontological elements in it. Schroedinger started his
theory on a number of principles: 1) that it should make contact with
classical physics somewhere, he chose the principle of energy
conservation. 2) That there are two fields involved, the
electromagnetic field and a \psi field associated with the wave
aspects of "particles" whose properties are to be determined by other
principles. 3) The Principle of Analogy requires that the \psi field
satisfy a second-order partial differential equation. 4) That the
structure of this differential equation be constrained by those
empirical principles that suggest a wave-particle duality, namely the
de Broglie-Einstein relations. The resulting Schroedinger equation is
one of the most brilliantly conceived equations in all of physics.

On the one hand, the Schroedinger equation is a hodgepodge of
principles. But, on the other hand, remember that a principle is a
heuristic or law in which you have strong confidence. So, if the
Schroedinger theory has been put together with inner consistency, its
empirical conclusions should stand the test of time, and as Einstein
put it, it will have "logical perfection and security of the
foundations."

I remember quite well one conversation I had with my mechanics
professor in 1975. I asked him what he thought would be the future of
physics. He was very convinced of two things: That classical
thermodynamics would remain, but that quantum mechanics would be
radically changed. At that time, I was a "good" physics students that
didn't get involved in all that subversive stuff of the philosophy of
physics or even Einstein's essays. So I had no perspective to
challenge him on his assessment of the future of physics. But today, I
have to chuckle at his reply. How ironic. Classical thermodynamics and
QM are both principle theories, so why should one stand the test of
time, but the other not?! I'm not referring, of course, to the future
providing "deeper explanations." He knew that classical thermodynamics
already had a deeper explanation in terms of kinetic theory. But if it
isn't an issue of the future providing a deeper explanation of QM,
then why wouldn't QM stand the test of time as well as classical
thermodynamics? Which principle of QM that I mentioned above is the
most likely to fail in the future? Perhaps the professor merely
stumbled into making a "false comparison" fallacy. That is, he set up
an analogy but didn't follow it properly. In any case, if physicists
were more conversant in the philosophy of physics perhaps they
wouldn't get these kinds of "extemporaneous philosophizings" goofed
up.

If QM is to fail, which principle that it is based on is going to be
the one that fails it?

Patrick

(Patrick Reany) wrote:
If QM is to fail, which principle that it is based on is going to be
the one that fails it?

That's a good question. The closest thing there is to an overriding
principle in the sense Einstein articulated in the (deleted) quote
is that there is a minimum uncontrollable dispersion for any system
in any state of any quantity that mixes velocities and coordinates.

The canonical quantization scheme is a way to realise this basic
assertion, but is not necessarily fundamental. A quantum system
might not have a classical limit whose inverse is described by
the canonical quantization scheme.

What is true, however, is that if the dispersion is described by
the non-commutativity of the variables involved; particularly:
u(q(t)) (u(q(t+dt))) - u(q(t+dt)) u(q(t)) = i h-bar W_uu(t) dt
then the basic assertion is that:
W_uu is non-zero, for any non-constant function u(q)
of the system's coordinates q(t)=(q1(t),q2(t),q3(t),...).

If you couple this with the assertion that:
q(t) satisfies equations of motion of the 2nd order:
q''(t) = a(q(t), q'(t)),
then it actually FOLLOWS (not assumed, but follows!) that the
system's classical limit is variables and equations of motion
described by a non-singular Lagrangian (and therefore a Hamiltonian)
and Poisson Brackets {,} satisfying the limit relation:
(FG - GF)/(i h-bar) - {F,G}.

So, the assertion that a quantum system be a quantization of a
Hamiltonian system is NOT required as a postulate, but is a theorem.

So, the basic axioms a
(1) The dispersion relation; W_uu required to be non-zero,
(2) The existence of 2nd order equations of motion.
And the only one that differentiates quantum from classical
mechanics is the assertion that W_uu be non-zero.

Note, that the general dispersion functional
W_uv (def)= (u dv/dt - dv/dt u)/(i h-bar)
is symmetric. Functions of the coordinates commute, by
axiom 1 (taking the limit dt - 0), so
uv - vu = 0.
Taking time derivatives:
(du/dt v - v du/dt) + (u dv/dt - dv/dt u) = 0
or
W_uv - W_vu = 0.
Thus, W_uv = W_vu.

If W_uu is 0 for all coordinates, then so are all the W_uv. This
follows since
0 = W_{u+v u+v} = W_uu + W_uv + W_vu + W_vv
= 2 W_uv.

If W_{qi qi} = 0 for coordinates qi, i = 1,2,...,M, then the entire
sector commutes: W_{qi qj} = 0, for all i, j = 1,2,...,M. So,
absent the first postulate, the system splits off into a classical
sector, given by purely classical coordinates, and a quantum
sector, given by coordinates qi, i = M+1,M+2,..., that don't
commute with their velocities.

The reason you don't want the classical sector emerges when it
comes time to try and establish a notion of statistical and
thermal physics on top of this foundation.

The variables q1,q2,...,qM in the classical sector can have an
arbitrarily small dispersion for a given state of the system.
For pure states, the dispersion of these q's will be 0. So,
when it comes time to tabulate the entropy of the system,
invariably you get to the question: how many (pure) states
are there for a given macro state. Unless the macro state is,
likewise, pure, the answer is infinity. The entropy is the
logarithm of the number of pure states the system can be in,
in a given macro state.

Put this another way: since the coordinates in this sector
have an arbitrarily small dispersion, then the amount of
information you can use the system to encode (in these
coordinates) is infinite. But, no physical system has an
infinite capacity for information. That would be
tantamount to having omniscience in a finite mortal body.
A system with an infinite information capacity can faithfully
encode the entire universe (even including itself).

Put this yet another way: the entropy of a system measures
the amount of information contained in the system's (pure)
micro state, for the system in a given macro state. A
system with an infinite entropy then contains an infinite
amount of information.

So, posing the more basic requirement that pure states not
be dispersion-free excludes the existence of a classical
sector and leads to the first postulate.

So, to ultimately answer your question: the most fundamental
assertion is that physical systems have a finite information
capacity, and that a finite system have a finite entropy
and have only a finite number of pure states available to it,
for any macro state.

So, the basic axioms a
(1) The dispersion relation; W_uu required to be non-zero,
(2) The existence of 2nd order equations of motion.

Note, that the general dispersion functional
W_uv (def)= (u dv/dt - dv/dt u)/(i h-bar)
is symmetric. Functions of the coordinates commute, by
axiom 1 (taking the limit dt - 0)

.... which I didn't explicitly list;

(1) The dispersion relation:
u(t) v(t+dt) - v(t+dt) u(t) = O(dt).
with
W_uv defined by (u(t)v'(t) - v'(t)u(t))/(i h-bar),
and
W_uu != 0 unless u(q) = constant.

As mentioned in the quoted article; it follows from (1) and
(2) that the classical limit of the system be that described
by a Hamiltonian H(q,v) with
(A(q,v)B(q,v)-B(q,v)A(q,v))/(i h-bar) - {A,B}
in the limit; and
dA/dt - {A,H}
in the limit.

The extra assertion required to characterize when the inverse
limit is described by Canonical quantization, I believe is:
(3) W_{qi qj} commutes with everyone for each i,j;
i.e., that the dispersion matrix W^{ij} = W_{qi qj} consist
of c-numbers only.

(Patrick Reany) wrote:
Einstein told us that he was once an etherist. He (apparently)
accepted the Mechanical Program's dogma to explain all phenomena in
terms of mechanics? But what is mechanics? Mechanics is the science
that MODELS all matter as being aggregates of point mass particles in
motion. Newton's mechanics adds to that description that masses
interact via forces acting-at-a-distance. (Hertz's mechanics --
inspired by his positivist leanings -- does not use the force concept
as a primitive.) The logical extension to this rigid approach would be
to approximate matter as a continuum state. The mathematical
distinction between these two approaches was profound, for in the
former case one uses total differential equations and in the latter
one uses partial differential equations. Einstein told us that this
led the way for the development of field theories of electrodynamics.

It's possible to establish continuum mechanics as a 'principle'
theory too. One starts with the assertion that for a perfectly
cohesive system the discrete (Newtonian) laws apply, particularly:
d(mv^2/2)/dt = F.v
().() denoting the dot product of vectors, and that this relation
pertain even in the smallest elements, so that if a system is
described by a continuum with
m_V = integral_V rho dV;
m_V = mass contained in volume V
and with
Perfect Cohesion:
integral_V rho f(x,v) dV = m f(x,v)
then
d/dt integral_{V(t)} rho v^2/2 dV = integral_{V(t)} rho/m F.v dV
over any "comoving" volume V(t) moving with.

This requires a prior conception of the mass continuum being
associated with co-moving volumes (i.e., volumes that flow along
the streamlines of the velocity field v(t)). From the transport
theorem, one then derives the transport equations under the
assumption of perfect cohesion:
@(rho v^2/2) + del.(v rho v^2/2) = rho b.v
@ denotes in ASCII the curly partial derivative symbol
where b = F/m is the commonly used symbol to denote the force
per unit mass.

That, then, is the starting point. One has a prior discrete
theory (he Newtonian mechanics) and from this arrives at
the idealized continuum form, as above.

Now, continuum mechanics, itself, follows by imposing two
invariance conditions:

(0) Correspondence Limit
In the limit of perfect cohesiveness, the equations of
motion are those derived from the corresponding discrete
mechanics.

(1) Scale Relativity or Additivity
The equations of motion be invariant with respect to
changes in level. In particular, if a system is
composed of parts
sum_a rho_a = rho
sum_a (rho_a v_a) = rho v
sum_a (rho_a b_a) = rho b
then the equations of motion satisfied by the system
as a whole are the same in form as those satisfied by
each of the parts.

(2) Galilean Relativity
The equations of motion are invariant with respect to
changes in the frame of reference.

Additionally, one can distinguish fundamentally continuous
systems from atomic systems by the assertion:

(3) Atomic Hypothesis
These exists a decomposition of a system into cohesive
subsystems.

But this assertion is immaterial for what follows.

Since the continuum equations arrived at by (0):
@(rho v^2/2)/@t + v.del(rho v^2/2) = rho b.v
are non-linear, and since non-linear combinations generally
do not preserve their form under summation of subsystems, these
equations are not scale invariant. This, then, requires
off-setting components which, if properly selected, will
also be invariant under changes in observer motion.

This leads to the definitions of quantities Tij, Wijk, qij
to respectively offset combinations of the form rho vi vj,
rho vi vj vk, rho vi bj, where v = (v1,v2,v3), b = (b1,b2,b3).
Then, one writes scale-invariant forms for the summations:

sum (rho_a vi_a vj_a + Tij_a) = rho vi vj + Tij

sum (rho_a vi_a vj_a vk_a + Tij_a vk_a + Tik_a vj_a + Tjk_a vi_a + Wijk_a)
= rho vi vj vk + Tij vk + Tik vj + Tjk vi + Wijk
and
sum (rho_a vi_a bj_a + qij_a) = rho vi bj + qij.

The quantities if interest in the equation of motion are
rho v^2/2, rho v v^2/2, and rho v.b.
The corresponding scale-invariant form for rho v^2/2 is then:
rho d_ij v^i v^j/2 + d_ij Tij/2
= rho (d_ij v^i v^j/2 + e)
= rho (v^2/2 + e)
where
e = trace(T)/(2 rho).
The summation convention is used above, and d_ij stands for the
Kroenecker Delta (in ASCII form).

For rho v.b, one has
rho d_ij v^i b^j + d_ij q^ij = rho v.b + q
where
q = trace(q).

And for rho v v^2/2, one has
1/2 rho d_jk v^i v^j v^k
= (rho d_jk v^i v^j v^k + d_jk (Tij v^k + Tik v^j + Tjk v^i) + d_jk Wijk)/2
= rho v^i (v^2/2 + e) + (T.v)^i + w^i
where
w^i = 1/2 d_jk Wijk = 1/2 Tr_{23}(W)
and
(T.v)^i = d_jk Tij v^k.

The equations of motion then assume the scale-invariant form:

@(rho (v^2/2 + e))/@t + del.(v rho (v^2/2 + e) + T.v + w) = rho b.v + q.

Under a change in frame of reference to an observe travelling at
a speed u relative to the original frame, one has the transformation:
v - u + v; @/@t - @/@t - u.del; del - del; rho - rho.

If F - F' under the transformation, then the combination
@F/@t + del.(v F)
transforms to:
@F'/@t - u.del F' + del.((u + v) F') = @F'/@t + del.(v F').

The particular forms chosen for the off-setting quantities,
T, W, q ensure that these are each invariant under Galilean
transformations. Therefore, the original equations transform to:

@(rho (|u+v|^2/2 + e)/@t +
del.(v rho (|u+v|^2/2 + e) + T.(u+v) + w) = rho b.(u+v) + q.

Subtracting out the original equation of motion, one gets:

@(rho (u^2/2 + u.v))/@t + del.(v rho (u^2/2 + u.v) + T.u) = rho b.u.

This split into a part linear in u, and a part quadratic in u,
which each must separately be equated:

@(rho u.v)/@t + del.(v rho (u.v) + T.u) = rho b.u
and
@(rho u^2/2)/@t + del.(v rho u^2/2) = 0.

From the first equation, factoring out the u.(), one gets:
@(rho v)/@t + del.(rho v v + T) = rho b
where
(rho v v + T) is the tensor with components (rho v^i v^j + Tij).
From the second equation, factoring out the u^2/2, one gets:
@rho/@t + del.(rho v) = 0.

The quantity T is identitied as the stress tensor of the system,
and e as the system's internal energy. They are related by:
1/2 trace(T) = rho e.

In fact, this is true for monoatomic gases. For diatomic and
polyatomic gases, there is an additional component to e that
does not arise from the 1/2 the trace of T.

(Patrick Reany) wrote:
Schroedinger started his theory on a number of principles: 1) that
it should make contact with classical physics somewhere, he chose
the principle of energy conservation. 2) That there are two fields
involved, the electromagnetic field and a \psi field associated with
the wave aspects of "particles" whose properties are to be determined
by other principles.

There's a major misconception (a fatal one) that needs to be cleared
up here. \psi is, properly speaking, not a field at all. For
a one particle system, you can think of it that way. But in
general, it describes the state of the WHOLE system, so that it's
a function
psi(q1,q2,q3,...,qN)
if the SYSTEM's confinguration space coordinates (q1,q2,...,qN);
not spacetime coordinates (x,y,z).

This should lead you immediately to the notion that psi is
not the fundamental object of consideration at all. Its
remoteness from any concept related to Classical mechanics
should likewise point you that way.

The von Neumann axiomatization revealed the more fundamental
concept of "state" underlying this, also a definition of
"state" that transcends the distinction between classical and
quantum physics. In the process, one also finds a distinction
between "pure" and "mixed" states. The latter cannot be
represented by psi's at all; so that even less can psi be
considered fundamental since it's not generally characteristic
of the general concept of state.

On the deeper foundation, you can ultimately PROVE that psi
has to take the form of a Hilbert space vector, for pure states.
This is on the assumption that the underlying algebraic system
describing the dynamical variables of the system be non-commutative
and satisfy the axioms that define a C*-algebra.

It is a theorem that the state space of a C*-algebra comprise the
density matrices over a Hilbert space, and that the pure states
be characterized (modulo norm) by non-zero Hilbert space vectors;
the psi's.

The more fundamental notion is not the psi's at all; but the
more basic conception which is shared by both classical and
quantum mechanics (resolving the other problem of remoteness,
in the process): the system's configuration space coordinates.

The commonality of classical and quantum mechanics is nearly
complete in this respect, so that nothing essentially new need
be posed at the outset in this regard. What IS new, in contrast,
is the basic assertion concerning the nature of these quantities:
namely, that they be non-commutative and satisfy a suitable set
of Heisenberg relations.

On the one hand, the Schroedinger equation is a hodgepodge of
principles. But, on the other hand, remember that a principle is a
heuristic or law in which you have strong confidence.

That's precisely the impression that result from being one step
removed above the more fundamental level, as described above.

"Wave functions" are a derivative concept, not basic to the
enterprise; and their presence is contingent on a LARGE set
of assumptions that aren't characteristic of the eseence of
QM either. They don't come first, but much later far down
the totem pole.

(Patrick Reany) wrote in message m...
(Daniel Weston) wrote in message ...
Patrick, you keep saying that QM is a principle theory, not a
constructive theory.
Could you explain this conclusion. My interpretation of Einstein's
comments that you have given us, is quite different. Thanks.

************************************************** ************

Is QM a principle theory?


If QM is to fail, which principle that it is based on is going to be
the one that fails it?

Patrick

Sorry for interrupting.

If Gr is connected to QM what has been accomplished?

Very interesting post. Probability statistics? If uncertainty is said
to exist, and probabilty diagrams allow for this to be answered, what
value would we find , when energy becomes specific, and identifies
each particle?

Strings then, would have dismissed uncertainty as well?

I am trying to digest the whole of your post. The idea here seems to
be a interplay between one position and another, and I find this
throughout my studies. A philosphical position first materializes
through the logic, and then comes the math? If a generalization is
spoken, and philopshically it is correct, then so is the math behind
it?

Topological Formation from Induction and Deduction:
(http://www.superstringtheory.com/for...ges10/321.html)

Sol said, "What I told you about quantum mechanics, and the
statistical probabilities, are the diverse ideas and experiences, that
are moved forward in democratic institututions. Nash understood the
negotiation process, and recognize the diversity in patterns, and came
up to a recognizable process that is now at the forefront of economic
functions in negotiations.

How quaint indeed that such simplicity is the rule in Feynman's
diagrams that we indeed understand complex issues now?"


Preparation for First
Principleshttp://superstringtheory.com/forum/d...ages10/66.html)



The Continuum Hypothesis:
(http://superstringtheory.com/forum/g...ages2/117.html)



Sol

(Patrick Reany) wrote in message m...
(Daniel Weston) wrote in message ...
Patrick, you keep saying that QM is a principle theory, not a
constructive theory.
Could you explain this conclusion. My interpretation of Einstein's
comments that you have given us, is quite different. Thanks.

************************************************** ************

Is QM a principle theory?

In the first place, all classifications are arbitrary. In the second
place, I know that some posters feel that the "principled vs
constructive" distinction serves no useful purpose in physics. I
disagree and I'll make clear exactly why I disagree later. The central
point of contention in the distinction between principle or
constructive theories is the existence of a "hypothetical" ontological
element in the theory. What is "hypothetical" and what is not as a
physical concept is time dependent and established by convention
(Poincare) of the physics community. Einstein's original idea of a
hypothetical ontological element was this: If it's assumed to exist,
but invisible, it's hypothetical. However, as useful as this notion
was a 100 years ago, it's too impractical now. For practicality,
instead of saying the "hypothetical electron" we just say the
"electron," and hopefully remember that electrons are hypothetical in
some ontological sense.

I have chosen the term "ontological element" carefully. By "ontology"
I mean existence in a material sense. To Einstein, the concepts of the
luminiferous ether and the atom, if they should correspond to anything
"real" at all, correspond to things materially "real," that is, they
correspond to something made of matter. Ordinary matter, whatever it
is metaphysically, is well describable physically: It has mass (get
enough of it and you can weigh it); it has inertial properties; it is
affected by the gravitational fields of other matter particles. Maybe
you can think of more. OK, enough introduction.

Let's begin with a quote from Einstein:

We can distinguish various kinds of theories
in physics. Most of them are constructive.
They attempt to build up a picture of the more
complex phenomena out of the materials of a
relatively simple formal scheme from which
they start out. Thus the kinetic theory of gases
seeks to reduce mechanical, thermal, and
diffusional processes to movements of molecules
-- i.e., to build them up out of the hypothesis of
molecular motion. When we say that we have
succeeded in understanding a group of natural
processes we invariably mean that a constructive
theory has been found which covers the
processes in question.
Along with this most important class of
theories there exists a second, which I will
call 'principle-theories'; These employ the
analytic, not the synthetic, method. The elements
which form their bases and starting-point are not
hypothetically constructed but empirically
discovered ones, general characteristics of
natural processes, principles that give rise to
mathematically formulated criteria which these
separate processes or the theoretical
representations of them have to satisfy. Thus
the science of thermodynamics seeks by
analytical means to deduce necessary conditions,
which separate events have to satisfy, from the
universally experienced fact that perpetual
motion is impossible.
The advantages of the constructive theory
are completeness, adaptability, and clearness,
those of the principle theory are logical
perfection and security of the foundations.
The theory of relativity belongs to the latter
class. In order to grasp its nature, one needs
first of all to become acquainted with the
principles on which it is based. Before I go
into these, however, I must observe that the
theory of relativity resembles a building
consisting of two separate stories, the special
theory and the general theory. The special
theory, on which the general theory rests,
applies to all physical phenomena with the
exception of gravitation; the general theory
provides the law of gravitation and its relations
to the other forces of nature.
Found in: "What is the Theory of Relativity?",
Einstein, Ideas and Opinions, Three Rivers
Press, p. 228-9.

I have written much on this quote on my website. See

http://www.ajnpx.com/html/Science/Ho...long-haul.html

http://www.ajnpx.com/html/Einstein's-search-for-meaningful-unification-in-physics.html


I can at this point only do two things to clear up what Einstein meant
by "principle vs constructive theories": First, I will place his
characterizations of them in his time period for a context, providing
his insight into Newton's mechanics as a starting point. Second, I
will attempt to clarify what I think can be made into a doable and
useful definition of these two terms. But first a definition. A
"principle" is a law or heuristic in which one has great confidence.

What Einstein meant "principle" is the same thing
*every* mathemadork since *Euclid* said the
same thing:

To wit, Geometry has *something* to with "physics".
And ever since that time all non-mathematicians have
been telling "scientists":

MORON has something to do with "science".

(sol) wrote in message om...
(Patrick Reany) wrote in message m...
(Daniel Weston) wrote in message ...
Patrick, you keep saying that QM is a principle theory, not a
constructive theory.
Could you explain this conclusion. My interpretation of Einstein's
comments that you have given us, is quite different. Thanks.

************************************************** ************

Is QM a principle theory?


If QM is to fail, which principle that it is based on is going to be
the one that fails it?

Patrick

Sorry for interrupting.

If Gr is connected to QM what has been accomplished?

Very interesting post. Probability statistics? If uncertainty is said
to exist, and probabilty diagrams allow for this to be answered, what
value would we find , when energy becomes specific, and identifies
each particle?

Strings then, would have dismissed uncertainty as well?

I am trying to digest the whole of your post. The idea here seems to
be a interplay between one position and another, and I find this
throughout my studies. A philosphical position first materializes
through the logic, and then comes the math? If a generalization is
spoken, and philopshically it is correct, then so is the math behind
it?

I'm not sure what the exact question is, but the answer is probably
no. People start with a formal point of view that is constrained by
their philosophical preferences, such as whether or not to adhere to
logical economy, the PoR, flat spacetime, or whatever. These are
decisions that can't be claimed merely on the basis of empircism.


Fundamental ideas play the most essential role in forming
a physical theory. Books on physics are full of mathematical
formulae. But thought and ideas, not formulae, are the
beginning of every physical theory.
--- Einstein & Infeld, The Evolution of Physics,
Touchstone, 1938, p 277.


In order to CONSTRUCT a theory, it is not enough to
have a clear conception of the goal. One must also
have a FORMAL POINT OF VIEW which will sufficiently
restrict the unlimited variety of possibilities.
--- Einstein, Ideas and Opinions, The
fundaments of theoretical physics, p. 328,
emphasis mine.

Patrick

(Alfred Einstead) wrote in message . com...
(Patrick Reany) wrote:
(Patrick Reany) wrote:
Einstein told us that he was once an etherist. He (apparently)
accepted the Mechanical Program's dogma to explain all phenomena in
terms of mechanics? But what is mechanics? Mechanics is the science
that MODELS all matter as being aggregates of point mass particles in
motion. Newton's mechanics adds to that description that masses
interact via forces acting-at-a-distance. (Hertz's mechanics --
inspired by his positivist leanings -- does not use the force concept
as a primitive.) The logical extension to this rigid approach would be
to approximate matter as a continuum state. The mathematical
distinction between these two approaches was profound, for in the
former case one uses total differential equations and in the latter
one uses partial differential equations. Einstein told us that this
led the way for the development of field theories of electrodynamics.

It's possible to establish continuum mechanics as a 'principle'
theory too. One starts with the assertion that for a perfectly
cohesive system the discrete (Newtonian) laws apply, particularly:
d(mv^2/2)/dt = F.v

This is power...

().() denoting the dot product of vectors, and that this relation
pertain even in the smallest elements, so that if a system is
described by a continuum with
m_V = integral_V rho dV;
m_V = mass contained in volume V
and with
Perfect Cohesion:
integral_V rho f(x,v) dV = m f(x,v)
then
d/dt integral_{V(t)} rho v^2/2 dV = integral_{V(t)} rho/m F.v dV
over any "comoving" volume V(t) moving with.

Is there anyway to explain this simpler, maybe using a
quantum of power.

This requires a prior conception of the mass continuum being
associated with co-moving volumes (i.e., volumes that flow along
the streamlines of the velocity field v(t)). From the transport
theorem, one then derives the transport equations under the
assumption of perfect cohesion:
@(rho v^2/2) + del.(v rho v^2/2) = rho b.v
@ denotes in ASCII the curly partial derivative symbol
where b = F/m is the commonly used symbol to denote the force
per unit mass.

right, but why is "b" preferred above "a" for
acceleration (??).

That, then, is the starting point. One has a prior discrete
theory (he Newtonian mechanics) and from this arrives at
the idealized continuum form, as above.

Now, continuum mechanics, itself, follows by imposing two
invariance conditions:

(0) Correspondence Limit
In the limit of perfect cohesiveness, the equations of
motion are those derived from the corresponding discrete
mechanics.

(1) Scale Relativity or Additivity
The equations of motion be invariant with respect to
changes in level. In particular, if a system is
composed of parts
sum_a rho_a = rho
sum_a (rho_a v_a) = rho v
sum_a (rho_a b_a) = rho b
then the equations of motion satisfied by the system
as a whole are the same in form as those satisfied by
each of the parts.

I'm presuming rho is density...

(2) Galilean Relativity
The equations of motion are invariant with respect to
changes in the frame of reference.

Additionally, one can distinguish fundamentally continuous
systems from atomic systems by the assertion:

(3) Atomic Hypothesis
These exists a decomposition of a system into cohesive
subsystems.

But this assertion is immaterial for what follows.

Since the continuum equations arrived at by (0):

Where's (0)?

@(rho v^2/2)/@t + v.del(rho v^2/2) = rho b.v
are non-linear, and since non-linear combinations generally
do not preserve their form under summation of subsystems, these
equations are not scale invariant.

If I'm not mistaken, you're describing
turbulence, and in aeronautics the Reynolds
number is uded to scale?

This, then, requires
off-setting components which, if properly selected, will
also be invariant under changes in observer motion.

This leads to the definitions of quantities Tij, Wijk, qij
to respectively offset combinations of the form rho vi vj,
rho vi vj vk, rho vi bj, where v = (v1,v2,v3), b = (b1,b2,b3).
Then, one writes scale-invariant forms for the summations:

sum (rho_a vi_a vj_a + Tij_a) = rho vi vj + Tij

sum (rho_a vi_a vj_a vk_a + Tij_a vk_a + Tik_a vj_a + Tjk_a vi_a + Wijk_a)
= rho vi vj vk + Tij vk + Tik vj + Tjk vi + Wijk
and
sum (rho_a vi_a bj_a + qij_a) = rho vi bj + qij.

The quantities if interest in the equation of motion are
rho v^2/2, rho v v^2/2, and rho v.b.
The corresponding scale-invariant form for rho v^2/2 is then:
rho d_ij v^i v^j/2 + d_ij Tij/2
= rho (d_ij v^i v^j/2 + e)
= rho (v^2/2 + e)
where
e = trace(T)/(2 rho).
The summation convention is used above, and d_ij stands for the
Kroenecker Delta (in ASCII form).

For rho v.b, one has
rho d_ij v^i b^j + d_ij q^ij = rho v.b + q
where
q = trace(q).

And for rho v v^2/2, one has
1/2 rho d_jk v^i v^j v^k
= (rho d_jk v^i v^j v^k + d_jk (Tij v^k + Tik v^j + Tjk v^i) + d_jk Wijk)/2
= rho v^i (v^2/2 + e) + (T.v)^i + w^i
where
w^i = 1/2 d_jk Wijk = 1/2 Tr_{23}(W)
and
(T.v)^i = d_jk Tij v^k.

The equations of motion then assume the scale-invariant form:

@(rho (v^2/2 + e))/@t + del.(v rho (v^2/2 + e) + T.v + w) = rho b.v + q.

Under a change in frame of reference to an observe travelling at
a speed u relative to the original frame, one has the transformation:
v - u + v; @/@t - @/@t - u.del; del - del; rho - rho.

If F - F' under the transformation, then the combination
@F/@t + del.(v F)
transforms to:
@F'/@t - u.del F' + del.((u + v) F') = @F'/@t + del.(v F').

The particular forms chosen for the off-setting quantities,
T, W, q ensure that these are each invariant under Galilean
transformations. Therefore, the original equations transform to:

@(rho (|u+v|^2/2 + e)/@t +
del.(v rho (|u+v|^2/2 + e) + T.(u+v) + w) = rho b.(u+v) + q.

Subtracting out the original equation of motion, one gets:

@(rho (u^2/2 + u.v))/@t + del.(v rho (u^2/2 + u.v) + T.u) = rho b.u.

This split into a part linear in u, and a part quadratic in u,
which each must separately be equated:

@(rho u.v)/@t + del.(v rho (u.v) + T.u) = rho b.u
and
@(rho u^2/2)/@t + del.(v rho u^2/2) = 0.

From the first equation, factoring out the u.(), one gets:
@(rho v)/@t + del.(rho v v + T) = rho b
where
(rho v v + T) is the tensor with components (rho v^i v^j + Tij).
From the second equation, factoring out the u^2/2, one gets:
@rho/@t + del.(rho v) = 0.

The quantity T is identitied as the stress tensor of the system,
and e as the system's internal energy. They are related by:
1/2 trace(T) = rho e.

In fact, this is true for monoatomic gases. For diatomic and
polyatomic gases, there is an additional component to e that
does not arise from the 1/2 the trace of T.

Agreed, would you say these "additional components"
are antisymmetric and or nonorthogonal components
off the trace, where the trace T solves only mono's?

Ken S. Tucker

(Ken S. Tucker) wrote:
The quantity T is identitied as the stress tensor of the system,
and e as the system's internal energy. They are related by:
1/2 trace(T) = rho e.

In fact, this is true for monoatomic gases. For diatomic and
polyatomic gases, there is an additional component to e that
does not arise from the 1/2 the trace of T.

Agreed, would you say these "additional components"
are antisymmetric and or nonorthogonal components
off the trace, where the trace T solves only mono's?

e and trace(T) are invariant under Galilean transformations, and
under the "scale" or "additivity" transformation. So, the
difference (e - 1/2 trace(T)) persists at all levels: microscopic
and macroscopic, and is intrinsic.

So, if the continuum decomposes into fundamental subsystems (the
"particles") the difference (e - 1/2 trace(T)) for that
subsystem represents a source of energy arising from intrinsic
degrees of freedom OTHER than the translational degrees.

It's direct evidence of the existence of non-translational
degrees of freedom in particles (i.e., spin); and is entirely
non-classical in origin.

In fact, for polyatomic gases, the e WILL actually be
1/2 trace(T) at low enough temperatures; because the other
degrees of freedom are frozen out. For diatomic gases, it pushes
up to 5/6 trace(T) past a certain critical temperature, as the
other degrees of freedom "thaw" out; and for polyatomic gases,
up to trace(T).