This note is a continuation of Why Algebra? Rudolf Haag
Haag’s Assertions
Concerning the interpretation of quantum mechanics Haag noted that
“to this day, there remains some uneasiness about its status, some disagreement concerning its interpretation. This is not restricted to crackpots. Different camps of eminent scientists advance widely different opinions.”
It is good, at this point, to remember the advice of the philosopher, Bertrand Russell, namely that when experts do not agree on a given subject, then “no opinion can be regarded as certain by a non-expert”. Then Haag defines his position, and I will concentrate on just two points here:
The subject of physics is ‘nature’ and, whatever this means precisely, it is something beyond and apart from human knowledge.
In other words, physics should be formulated in such a way that it concerns objective phenomena. All twists, so popular, trying to make physics subjective, invoking the necessity of an “observer”, and so on, do not belong to physics! They result from arbitrarily assumed metaphysical prejudices that affect rationality of thinking.
Then we have the second assertion:
B. Individual knowledge is gained by observation, typically by experiment. In describing the set up and the result of an experiment we are bound by limits emphasized by Bohr: ‘We must be able to tell our friends what we have done and what we have learned.’ Bohr concludes from this that in the description of both the arrangement and the result we are bound to use “the language of classical physics’’.
Thus Haag takes a rational point of view. The language of physics, one that physicists should use when talking about Nature, should be a classical language that deals with real, material objects and their placement in space time.
Popper’s “Propensity”
Further in his paper he notes that many issues in quantum theory are still open, in particular the issue of whether the correct description should be deterministic or not. Instead of using the term “probability”, that requires repeated experiments, Haag opts for the Popperian term “propensity”, that characterizes an objective property associated with a given situation. The fact that “propensity” per se cannot be observed should not worry us, because:
The use of concepts in the theory which are not directly amenable to observation is neither forbidden nor unusual. It seems unavoidable.”
Then he addresses the quantum mechanical superposition principle and superselection rules:
Strict superselection rules forbid (coherent) linear combinations of states which differ in some charge quantum number (electric, baryonic, leptonic,…).
This alone, at least for me, dismisses the “Schrödinger cat paradox”; the paradox exists only because it is based on arbitrary assumptions.
The fact that there IS a paradox tells us that these assumptions are wrong. But, for some reason, most physicists simply ignore this fact.
The Problem of the Observer
The concept of an “observer” was, for Haag, and also for me, causing a serious problem, as it meant that we forcefully limited the scope of physics, which was/is unnecessary. Instead of introducing an “observer” that is external to the physical system, it is better to introduce the concept of an event – as a fundamental concept. Things happen, with or without observers. Reality is Real on its own terms. Another fundamental concept that we should not be afraid of is irreversibility at a fundamental level. All these corrections should lead to:
…a self consistent theory in which all relevant objects are included as parts of the physical system.
Then Haag discusses different possible strategies that could lead to such a theory, and, in particular, notes:
In a series of papers Ph. Blanchard and A. Jadczyk have described a formalism which generates real events in the interaction of an atomic object with a macroscopic measuring device. The latter is idealized as a classical system in the sense that it is described by a commutative algebra which may include some discrete variables. The scheme, called “Event Enhanced Quantum Theory”, introduces irreversible decisions into the interaction process and yields a good phenomenological description of the quantum measurement process. (Italics mine)
As this concerns my own contribution, it is time for me to develop the comment of my teacher, one of the clearest and sharpest minds I ever knew in my life.
P.S.1 My wife likes it. SC looks a lot like FLA - Her childhood home. This is for Her.
P.S.2.12-06 13:00 I am trying to find out if the conformally compactified Minkowski space is orientable or not. Its double cover certainly is (it is S^1 x S^3), but its image by the covering map? I feel really shameful that I don't know it. Big holes in my working knowledge of differential geometry. Work, work, work and more work needed! So I work.
Diana Walsh Pasulka
P.S.5. The beginning of "Geometry of the conformally compactified Minkowski space"
"Thus, Protoconsciousness can be imagined as current awareness by a living cell of the gap, that being expressed as non-congruence, divergence, incompatibility, collision, conflict - between the ideal geometrical form and its physical realization. This means that such discrepancy between the Ideal Geometry and Robust Physics is "felt" by the cell. The ideal geometry is species-specific, initially pre-existing, and pre-determined, while the robust physics is actually occurring and constantly fluctuating to adapt, to adjust, to fit, to accommodate, to approximate to the ideal geometry. Accordingly, the living process can be expressed as continuous dynamic approximation of the "real" physical form to its geometrical "ideal". Since this approximation is felt until there is the non-congruence between physical and geometrical (which can diminish only asymptotically, i.e. the physical will never coincide with geometrical), then the geometrical feeling (protoconsciousness) is an inalienable part of any living entity. Hence, the Geometrical Feeling is suggested for the role of the Protophenomenal Fundamental alongside physical fundamentals (Mass, Charge, Time/Space). As to the deep ontological meaning of the concept of "Geometrical Feeling", it could be analogized with the "Universal Grammar" by N. Chomsky (1988), which may be considered as "intrinsic part of the structure of matter ever since the Big Bang, or a necessary part of the eternal Platonic world oflogic and mathematics" (Hamad, 2001). "
P.S.7. From Christopher Langan "The Theory of Theories":
"For example, modern physics is bedeviled by paradoxes involving the origin and directionality of time, the collapse of the quantum wave function, quantum nonlocality, and the containment problem of cosmology. Were someone to present a simple, elegant theory resolving these paradoxes without sacrificing the benefits of existing theories, the resolutions would carry more weight than any number of predictions. Similarly, any theory and model conservatively resolving the self-inclusion paradoxes besetting the mathematical theory of sets, which underlies almost every other kind of mathematics, could demand acceptance on that basis alone. Wherever there is an intractable scientific or mathematical paradox, there is dire need of a theory and model to resolve it.
If such a theory and model exist – and for the sake of human knowledge, they had better exist – they use a logical metalanguage with sufficient expressive power to characterize and analyze the limitations of science and mathematics, and are therefore philosophical and metamathematical in nature. This is because no lower level of discourse is capable of uniting two disciplines that exclude each other‘s content as thoroughly as do science and mathematics.
Now here‘s the bottom line: such a theory and model do indeed exist "
I don't think so. Attempts to create such a beast - they exist.
P.S.8. In the pdf note I have made a few changes comparing to the previous version.. I have multiplied the factor in front of the matrix by "i", and the matrix by "-i". The whole matrix U(Z) is unchanged. The proof (I will write it later today) will be somewhat more "elegant" after these "cosmetical" changes.
Where did I get the matrix U(Z) from? I am not sure. Reading, thinking, making errors, and correcting them. Until it works the way I envisaged it to work.
P.S.9. Uploaded a new version of the pdf note. The proof of the last lemma is not yet finished.
"I am trying to find out if the conformally compactified Minkowski space is orientable or not. Its double cover certainly is (it is S^1 x S^3), but its image by the covering map?".
ReplyDeleteGiven that the conformally compactified Minkowski space is represented as the topological product S^1 x S^3 (which essentially is a 4D space where one dimension is time-like and the other three are space-like), it is indeed orientable. This is because both S^1 and S^3 are themselves orientable (an n-sphere S^n is orientable for all n), and the product of orientable manifolds is also orientable.
If the double cover is orientable, it doesn't necessarily mean that the original space is orientable. However, in this case, since the original space (conformally compactified Minkowski space) is orientable, its double cover is indeed orientable as well.
"It is a Lie group, so it should be orientable. But then I don't understand anything at all.".
ReplyDeleteAh, I see where the confusion is coming from. Indeed, it is known that all Lie groups are orientable. This is a consequence of the definition of a Lie group as a smooth manifold that is also a group in which the group operations are smooth. Because it is a differentiable manifold with a consistent tangent bundle, it can always be oriented.
The conformally compactified Minkowski space is often seen as a projective space, specifically the real projective space RP^4. Now, the confusion arises because RP^4, unlike lower-dimensional real projective spaces, is non-orientable.
However, there is a way out of this contradiction. What is crucial here is that, although the full conformal group of the Minkowski space is indeed a projective special orthogonal group (which is a Lie group and thus orientable), it acts on the Minkowski space not quite faithfully: some elements of the group represent the same transformation of the Minkowski space. These are exactly the transformations changing the orientation. So, when we pass to the quotient by these transformations (that is, effectively consider the orientation), we obtain the space that is not orientable, even though its group of symmetries still is.
Therefore, we see that the conformally compactified Minkowski space is not orientable, even though the group of its symmetries is. This sounds contradictory, but it is exactly the kind of situation where one needs to be careful with the definitions and not jump to conclusions based solely on the properties of the symmetry group!
Perhaps this is the solution? Or make your questions more specific.
It doesn't help much, but thank you for your good intentions. I will soon start writing it all in details.
DeleteIn a private communication a Reader asked these questions concerning my notes on the conformally compactified Minkowski space:
ReplyDelete"Why do you introduce exactly the Minkowski space? What is the philosophical basis? What do you want to describe and why? What is the question you are looking for an answer to when writing these calculations?"
These are good questions and I will address them at some point.
We have all kinds of dreams, ideas, intuitions. Some of them later prove to be good, some need to be discarded. These intuitions led us to some fields that are not yet plowed. So they need a plow and a ploughman. And a hard work. Plowing must go deep enough. Then, there is a chance that the fields will bear fruit, and that our new knowledge will bring us closer to the answer to our conscious and unconscious questions.
One way of thinking goes like this.
ReplyDeleteLife as we know it exists because of electromagnetism. Linear electromagnetism needs and implies conformal structure that implies the conformal group. Conformal group leads us to conformal compactifications.