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dextercioby submitted a new PF Insights post
The Need of Infinity in Physics
Continue reading the Original PF Insights Post.
The Need of Infinity in Physics
Continue reading the Original PF Insights Post.
Hmm. Don't they use the wave formulation a lot in chemistry and matter physics?bhobba said:Nice write up.
Indeed QM requires an infinite dimensional vector space. But I have to mention in modern times neither Martix Mechanics or Wave Mechanics are used. What's used is the more general transformation theory worked out by Dirac that incorporates both, plus his rather strange q numbers:
http://www.lajpe.org/may08/09_Carlos_Madrid.pdf
Thanks
Bill
bhobba said:Nice write up.
Indeed QM requires an infinite dimensional vector space. But I have to mention in modern times neither Martix Mechanics or Wave Mechanics are used.
Greg Bernhardt said:Interesting counter article
http://blogs.discovermagazine.com/crux/2015/02/20/infinity-ruining-physics/
dextercioby said:dextercioby submitted a new PF Insights post
The Need of Infinity in Physics
Continue reading the Original PF Insights Post.
There’s always the feeling you get when you study physics really deeply that you’re doing no more than applied mathematics.
Unless you’re a modern Michael Faraday, i.e. a guy who works in a team who works in a (sometimes really big) laboratory from a (typically huge) facility or research institute like CERN or FermiLab, and your day-to-day job involves working with electronic equipment.
Do let a Heisenberg matrix be finite (Avogadro’s number of lines and columns) and you won’t have a quantum theory whatsoever. [as a side note: do let Planck’s constant be = 0 and you won’t have a quantum theory again].
davidbenari said:Hmm. Don't they use the wave formulation a lot in chemistry and matter physics?
Dr. Courtney said:Maybe not in your end, but I recall writing and running codes to diagonalize really big matrices (100k by 100k) in atomic physics.
Tabasko633 said:Isn't one of the reasons that time and space are continuous? So we need the concept of infinity to calculate dynamics in such a environment?
Actually, the "size" (or cardinality) of an infinite set can change when you add an infinite number of elements to an infinite set (example: add a set of the size (cardinality) of ##\mathbb R## to a countable infinite set).mma said:But if you add infinite number of elements to an infinite set, then the "size" of the set remains unchanged. This is the difference.
You are right. I should have told countably infinite.Samy_A said:Actually, the "size" (or cardinality) of an infinite set can change when you add an infinite number of elements to an infinite set (example: add a set of the size (cardinality) of ##\mathbb R## to a countable infinite set).
Schrödinger’s operators for coordinate and momentum make sense only in an infinite dimensional Hilbert space, as a consequence of Stone-von Neumann’s theorem (1931).
Hermann Weyl observed that this commutation law was impossible to satisfy for linear operators p, x acting on finite-dimensional spaces unless ℏvanishes. This is apparent from taking the trace over both sides of the latter equation and using the relation Trace(AB) = Trace(BA); the left-hand side is zero, the right-hand side is non-zero. Further analysis[6] shows that, in fact, any two self-adjoint operators satisfying the above commutation relation cannot be both bounded.
What you say is very hazy and strange to me. Could you give a reference?bhobba said:Personally in QM I consider the physical realizable states to be finite dimensional, but perhaps of very large dimension, experimentally indistinguishable from an actual infinite one. One then, for mathematical convenience, and since we don't actually know the dimension, introduces states of actual infinite dimension so the powerful theorems of functional analysis can be used.
Shyan said:What you say is very hazy and strange to me. Could you give a reference?
bhobba said:The dual of all the row vectors of finite dimension contains everything used in QM - its in fact the maximal space of a Gelfland tripple:
https://en.wikipedia.org/wiki/Rigged_Hilbert_space
All the spaces used in QM are a subset of this space.
Thanks
Bill
Shyan said:I think what I actually don't understand is what definition of dimension you are using. Because for me dimension is a property of the space not the states!
"The "core" of all mathematics is the rigor of deductive logic applied to axiomatics"jambaugh said:" It is beyond doubt that the notion of infinity lies somewhere near the core of all mathematics..." I dispute vehemently. The "core" of all mathematics is the rigor of deductive logic applied to axiomatics. Infinities manifest when it is improper or inconvenient to impose the actual finiteness we find in applications of mathematics.
The notion of infinity should, in physics, always and only be understood as a place-holder for an unspecified finite boundary.
Physically we never actualize infinities except possibly in the measure of ignorance which is always infinite in contrast to our finite knowledge. The corollary to this is finite information encoded in an infinitude of possible ways which underlies the mysteries of quantum mechanics.
This is not to say that we should discard the (*mathematical*) concept. Many people measuring distances each using distinct minimal units of precision would not have their measurements readily comparable unless we mapped them all into the "infinite precision" ideal of a continuum of measurements. We, also, may extrapolate well beyond the effects of given conditions in a certain application model and when doing so it is convenient to speak of "behavior at infinity" but this is simply short hand for "behavior beyond the significant influence of the aforementioned effects."
In short Infinity = Ignorance (as to where the boundary lies in some application of the theory).
If some construct, (such as the continuum of space-time) is necessarily infinite then we should always second guess any attempt to treat such a construct as manifestly physical. (Hence, do NOT take too seriously the ontological reality of the "space-time" manifold and its geometry.)
Or so I would assert.
Shyan said:I think what I actually don't understand is what definition of dimension you are using. Because for me dimension is a property of the space not the states!
jambaugh said:" It is beyond doubt that the notion of infinity lies somewhere near the core of all mathematics..." I dispute vehemently. The "core" of all mathematics is the rigor of deductive logic applied to axiomatics. Infinities manifest when it is improper or inconvenient to impose the actual finiteness we find in applications of mathematics.
The notion of infinity should, in physics, always and only be understood as a place-holder for an unspecified finite boundary.
Physically we never actualize infinities except possibly in the measure of ignorance which is always infinite in contrast to our finite knowledge. The corollary to this is finite information encoded in an infinitude of possible ways which underlies the mysteries of quantum mechanics.
This is not to say that we should discard the (*mathematical*) concept. Many people measuring distances each using distinct minimal units of precision would not have their measurements readily comparable unless we mapped them all into the "infinite precision" ideal of a continuum of measurements. We, also, may extrapolate well beyond the effects of given conditions in a certain application model and when doing so it is convenient to speak of "behavior at infinity" but this is simply short hand for "behavior beyond the significant influence of the aforementioned effects."
In short Infinity = Ignorance (as to where the boundary lies in some application of the theory).
If some construct, (such as the continuum of space-time) is necessarily infinite then we should always second guess any attempt to treat such a construct as manifestly physical. (Hence, do NOT take too seriously the ontological reality of the "space-time" manifold and its geometry.)
Or so I would assert.
davidbenari said:Hmm. Don't they use the wave formulation a lot in chemistry and matter physics?
The Need of Infinity in Physics refers to the idea that certain physical phenomena, such as the behavior of subatomic particles and the structure of the universe, cannot be fully explained without considering the concept of infinity. This concept challenges traditional notions of finite quantities and finite physical laws, and has been a topic of debate among physicists for centuries.
Infinity is important in physics because it allows for a more complete understanding of certain phenomena that cannot be explained by finite concepts. For example, the concept of infinity is crucial in the study of quantum mechanics and cosmology, where it helps to explain the behavior of particles at the subatomic level and the expansion of the universe.
The concept of infinity affects our understanding of the universe by challenging our traditional notions of space, time, and matter. It allows us to consider the possibility of infinite universes and infinite dimensions, and to better understand the fundamental laws that govern the behavior of the universe.
While infinity is a useful concept in physics, there are limitations to its use. For example, it is often difficult to apply infinitesimal quantities to real-world situations, and the concept of infinity can lead to paradoxes and contradictions if not used carefully. Additionally, not all physical phenomena can be explained by invoking the concept of infinity.
The concept of infinity is being studied and applied in modern physics through various mathematical and theoretical frameworks. For example, the concept of infinity is essential in the study of fractals and chaos theory, and has also been incorporated into theories such as string theory and loop quantum gravity. Additionally, advances in technology have allowed for experiments and observations that further our understanding of the role of infinity in the physical world.