Difference between revisions of "Dimensions in Physics"

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	**Non-commutative Geometry**: Grothendieck's ideas have also influenced non-commutative geometry, a field that extends classical geometry to quantum spaces. This has applications in quantum field theory and string theory, where the geometry of space-time itself can exhibit quantum properties².		**Non-commutative Geometry**: Grothendieck's ideas have also influenced non-commutative geometry, a field that extends classical geometry to quantum spaces. This has applications in quantum field theory and string theory, where the geometry of space-time itself can exhibit quantum properties².
−	Andreas Döring, ''Spectral presheaves as quantum state spaces'' file:///C:/Users/rp_to/Downloads/d%C3%B6ring-2015-spectral-presheaves-as-quantum-state-spaces.pdf	+	Doring introduced the mathematical setting and the key physical question: what is a physically sensible and useful definition of a quantum state space?<ref>Andreas Döring, ''Spectral presheaves as quantum state spaces'' file:///C:/Users/rp_to/Downloads/d%C3%B6ring-2015-spectral-presheaves-as-quantum-state-spaces.pdf</ref><blockquote>For each quantum system described by an operator algebra A of physical quantities, we provide a (generalized) state space, notwithstanding the
		+	Kochen–Specker theorem. This quantum state space is the spectral presheaf Σ. We formulate the time evolution of quantum systems in terms of Hamiltonian flows on this generalized space and explain how the structure of the spectral presheaf
		+	Σ geometrically mirrors the double role played by self-adjoint operators in quantum theory, as quantum random variables and as generators of time evolution.</blockquote>

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Revision as of 20:06, 2 September 2024

Full Title or Meme

Beyond the 3 space dimensions and 1 time dimension other dimensions have been proposed to deal with issues not addressed in these.

Context

In understanding physics, dimensions play a crucial role in describing physical quantities.

The common use of calculus in Mathematics assumes continuity, the real world can surprise us with its discrete and varied dimensions. Physics bridges this gap by ensuring that equations remain dimensionally consistent, allowing us to explore the universe with mathematical precision.^[1]

1. Dimensional Analysis

  - The dimension of any physical quantity expresses its dependence on the base quantities (such as length, mass, time, etc.). Each dimension is represented by a symbol (or a power of a symbol) corresponding to the base quantity.
  - For instance:
    - Length has dimension **L** or **L^1**.
    - Mass has dimension **M** or **M^1**.
    - Time has dimension **T** or **T^1**.
  - We can express the dimension of any physical quantity as:
    \[ \text{Dimension} = L^a M^b T^c I^d \Theta^e N^f J^g \]
    where \(a\), \(b\), \(c\), \(d\), \(e\), \(f\), and \(g\) are powers associated with each base quantity.
  - Quantities with dimensions that can be written with all seven powers equal to zero (i.e., \(L^0 M^0 T^0 I^0 \Theta^0 N^0 J^0\)) are called **dimensionless** or **pure numbers**.

1. Continuity and Dimensionality

  - In mathematics, we often assume continuity. However, in the real world, physical dimensions need not always be continuous.
  - For example, consider motion. We postulate that motion is continuous, but this assumption doesn't necessarily hold in reality.
  - Physicists use square brackets around the symbol for a physical quantity to represent its dimensions. For instance:
    - If \(r\) represents the radius of a cylinder and \(h\) represents its height, we write \([r] = L\) and \([h] = L\) to indicate that both dimensions are lengths.
    - Surface area (\(A\)) has dimensions \(L^2\), and volume (\(V\)) has dimensions \(L^3\).
    - Mass (\(m\)) has dimensions \(M\), and density (\(\rho\)) has dimensions \(M/L^3\).#. Dimensional Consistency
  - Any mathematical equation involving physical quantities must be dimensionally consistent.
  - Rules for dimensional consistency:
    - Every term in an expression must have the same dimensions.
    - Adding or subtracting quantities of differing dimensions is nonsensical (like adding apples and oranges).
    - Equations must have the same dimensions on both sides.

(1) 1.4 Dimensional Analysis | University Physics Volume 1 - Lumen Learning. https://courses.lumenlearning.com/suny-osuniversityphysics/chapter/1-4-dimensional-analysis/.

(2) 2.2: Units and dimensions - Physics LibreTexts. https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_Introductory_Physics_-_Building_Models_to_Describe_Our_World_%28Martin_Neary_Rinaldo_and_Woodman%29/02%3A_Comparing_Model_and_Experiment/2.02%3A_Units_and_dimensions.

(3) 6.1: An Analytic Definition of Continuity - Mathematics LibreTexts. https://bing.com/search?q=do+all+physical+dimensions+need+to+be+continuous.

(4) 1.7: Limits, Continuity, and Differentiability. https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_%28Boelkins_et_al.%29/01%3A_Understanding_the_Derivative/1.07%3A_Limits_Continuity_and_Differentiability.

Kaluza–Klein is a classical unified field theory of gravitation and electromagnetism built around the idea of a fifth dimension beyond the common 4D of space and time and considered an important precursor to string theory. In their setup, the vacuum has the usual 3 dimensions of space and one dimension of time but with another microscopic extra spatial dimension in the shape of a tiny circle. Gunnar Nordström had an earlier, similar idea. But in that case, a fifth component was added to the electromagnetic vector potential, representing the Newtonian gravitational potential, and writing the Maxwell equations in five dimensions.

Problems

1. For the Schrödinger equation to work, the space is typically defined as only piece-wise continuous, but a single photon may travel for a billion years or more, which seems to imply that all of 4-space must be continuously differentiable, not just in little patches.
2. Spooky Action at a Distance was what Einstein called non-locality in QM. Here we will look at violations of Bell's Inequalities are summarized nicely in popularizations by Anton Zeilinger^[2] and in the Scientific American^[3] which shows how Anton Zeilinger and others were able to describe quantum features like complementary pairs that are still not deterministic until observed.
3. Dark Matter seems to be more prevalent than Matter that is sensible to humans as real. Since more is not sensible than sensible, it would make more sense to just assume that no Matter is directly sensible to humans, but just the effects of that matter that is associated with the Schrödinger equation.
4. Dark Energy is more nebulous even than Dark Matter. Dark energy is an unknown form of energy that pervades the cosmos. Its primary effect is to drive the accelerating expansion of the universe. Using the lambda-CDM model of cosmology, dark energy constitutes 68% of the total energy in the observable universe today. But the above item deals with Dark Matter and Einstein tells us that if c = 1 then E = m. Therefore i conclude that the thing driving the universe apart is not energy conservation, but momentum conservation. What is also interesting that if the density of momentum (along with everything else in the universe) is tending towards zero density. That really sounds a lot like the second law to me.
5. The quantization of energy levels appears both in quantum and classical mechanics, and it is not a consequence of the Schrödinger equation. It is a consequence of confinement. In fact, anytime that a wave equation (any quantum equation for the wave-function, or a classical equation for a classical field, e.g., EM field) has periodic Boundary Conditions in some spatial variables, the system exhibits quantized energy levels. Unbound systems do not exhibit any quantized behavior.

Modeling

"Mathematical models power our civilization—but they have limits." Models are not reality. As we humans try to model ever more complex part of reality, like fluid flow and brain surgery, we inevitably come to real world phenomena that the model fails to predict.^[4] In those cases it is a common practice to switch to discrete or perturbation techniques.

It’s hard for a neurosurgeon to navigate a brain. A key challenge is gooeyness. The brain is immersed in cerebrospinal fluid; when a surgeon opens the skull, pressure is released, and parts of the brain surge up toward the exit while gravity starts pulling others down. This can happen with special force if a tumor has rendered the skull overstuffed. A brain can shift by as much as an inch during a typical neurosurgery, and surgeons, who plan their routes with precision, can struggle as the territory moves.

The Three Spacial Dimensions

There is no Matter/Energy or Momentum in these dimension, only the information contained in the Schrödinger equations occupies space. That is not to say the the effects of the collapse of the wave equation is not noticeable in 3-space, only that they are only visible as a part of that collapse. All we see are the observables, that should be obvious by its name. The only reason that continuity is a good description of these dimensions, is that there are no individual pieces (parts) in them, not matter/energy, not momentum.

The Fourth Dimension - Time

The conjugate value to time in Quantum Mechanics is energy, which is supposed to be the conserved quantity from Nöther's Theorem.

What i would like to consider is that the conserved quantity corresponding to time is information.

Types of Information

• Laplace - the clockwork universe is set in motion and runs deterministically by itself (Laplace's daemon)
• Gaussian - error bounds prevent us from make any exact measurement
• Boltzmann statistics of distinguishable (known) particles
• Planck & Bose - statistics of indistinguishable particles
• Pauli & Dirac - statistics of particles subject to exclusion principle
• Fisher - population statistics
• Shannon - information content

Entropy

The entropy of the universe is constantly increasing. The second law of thermodynamics states that the state of entropy of the entire universe, as an isolated system, will always increase over time. Energy always flows downhill, and this causes an increase of entropy. Entropy is the spreading out of energy, and energy tends to spread out as much as possible. It flows spontaneously from a hot (i.e. highly energetic) region to a cold (less energetic) region. As a result, energy becomes evenly distributed across the two regions, and the temperature of the two regions becomes equal. The same thing happens on a much larger scale. The Sun and every other star are radiating energy into the universe. However, they can’t do it forever. Eventually the stars will cool down, and heat will have spread out so much that there won’t be warmer objects and cooler objects. Everything will be the same very cold temperature. Once everything is at the same temperature, there’s no reason for anything to change what it’s doing. The universe will have run down completely, and the entropy of the universe will be as high as it is ever going to get.^[5]

The change in entropy of an isolated system during an irreversible process is > 0; while for a reversible process, it is = 0.the change in entropy of an isolated system during an irreversible process is > 0; while for a reversible process, it is = 0.

The collapse of the wave function is often considered to be irreversible. This transition is theoretically reversible. However, for practical purposes, it is effectively irreversible due to the interaction with the environment.

Requirement for a Metric

Not every dimension necessarily has a metric. In mathematics, a metric is a function that defines a distance between elements of a set. For a space to have a metric, it must satisfy certain properties like non-negativity, identity of indiscernibles, symmetry, and the triangle inequality.

However, there are many types of spaces in mathematics, and not all of them are metric spaces. For example, topological spaces are more general and do not require a metric to define their structure. Instead, they rely on open sets to describe the space’s properties. In a topological space, the concept of “closeness” or “continuity” is defined using open sets rather than distances. This allows for a broader range of spaces to be studied, including those that cannot be described by a metric.

For example, in a topological space, we can talk about concepts like convergence, continuity, and compactness without needing a specific distance function. This flexibility makes topology a powerful tool in many areas of mathematics.

Quantum Entanglement

The role of topology and its ability to preserve properties can be likened to how a coffee mug can be reshaped into the form of a doughnut; despite the changes in appearance and shape during the transformation, a singular hole – a topological characteristic – remains constant and unaltered. In this way, the two objects are topologically equivalent. “The entanglement between our photons is malleable, like clay in a potter’s hands, but during the molding process, some features are retained”. The nature of the topology investigated here, termed Skyrmion topology, was initially explored by Tony Skyrme in the 1980s as field configurations displaying particle-like characteristics. In this context, topology refers to a global property of the fields, akin to a piece of fabric (the wave function) whose texture (the topology) remains unchanged regardless of the direction in which it is pushed.^[6]

Condensed Matter Physics

Topological concepts are used to classify different phases of matter, known as topological phases. These phases have unique properties that remain invariant under continuous deformations. For example, topological insulators are materials that conduct electricity on their surface but act as insulators in their bulk.^[7]

Quantum Field Theory

In quantum field theory, topological spaces help describe the behavior of fields and particles. The concept of topological invariants is crucial in understanding phenomena like the quantum Hall effect, where the conductivity is quantized and remains constant despite changes in the material’s shape or size.

General Relativity

Topology is used to study the global properties of spacetime. For instance, the topology of a black hole’s event horizon can provide insights into the nature of singularities and the structure of the universe2.

String Theory

In string theory, the different possible shapes of extra dimensions (known as Calabi-Yau manifolds) are studied using topological methods. These shapes determine the physical properties of the universe at very small scales2.

Postulates

This is mostly an attempt to come up with a common taxonomy so there will be no confusion about what is said below

• Reality is limited to what humans together with their machine aids can directly sense in their normal world (called sensible reality below.)
• Nöther's Theorem should have the taxonomy revised like so. (We will describe the things conserved a little differently below.)

  For every continuous symmetry of the action of a system with conservative force has a corresponding dimension where conservation is maintained.

Solution

Non-locality

The following will shown one way to get the right answer for the split photon paradox as describe by Anton Zeilinger in the problem of non-locality as described above with a simple inversion of what we count as reality.

• The accepted 4-space of Minkowski is limited to passing the information wave packet known as the Schrödinger equation. No Matter/Energy or Momentum exist in these dimensions. This is the space that provides all of the input to our sensations of reality.
• There are (at least) two more dimensions that need to be considered; I will call them Energy and Momentum. While this is suggestive of my thoughts, I want to be clear that we could easily pick other names for them. The two are associated with the things that are not (always) a part of our sensible reality, which is limited to 4-space.
• These dimensions should not be considered spacial in the sense of the first three in that continuity is not present and the metric might be something we have not considered before.
  • If a geometry has an associate distance function, called a metric, then according to Ostrowski's theorem, the metric is essentially locally Euclidean or fractal, represented by the real numbers or the p-adic numbers, respectively.
  • Going with the idea that God created the integers and nothing else in Mathematics, I will stick with integers as the only arithmetic metric. If Geometry is used triangles have the advantage of being completely rigid.
  • In the Emperor's New Mind Penrose speculated on the link between undecidability and fractal geometry.
• The ensemble interpretation of quantum mechanics considers the quantum state description to apply only to an ensemble of similarly prepared systems, rather than supposing that it exhaustively represents an individual physical system.
• If some measured quantity, like frequency or its cognate energy, are created in a continuum then integers will not be able to capture that quantity. Computer science attempt to capture such continuous measurements as a floating point number in two parts, the value (aka fraction) and the exponent. Its accuracy is determined by the size of the value field, its range is determine by the size of the exponent field.
• When we make rules concerning events that include these dimensions we accept one or more entries from the list and create one or more entries at the end of the event. Just imagine this like a Feynman diagram. An alternate view would be a Ruliad like that described by Wolfram.^[8] The idea that their are rules that are applied in sequence can also be viewed as the origin of time itself.
• If we assume that there is no way to determine what will happen without running the rules to see the result, then an exact prediction of the future state of the universe is not possible. The only way to see the future in complete detail is the run the rules of the universe to see what happens. That implies that we cannot create a set of equations which tells us the exact future. In other words, if Quantum Mechanics is discrete, it cannot be predicted exactly with continuous functions.
• Conservation of energy and momentum means that the total values input of these two quantities will also be identical in the sum of the values in the output entries.

Dimension with No Exclusion

In particular a dimension where the Las of the

• Point Set Topology

Alexander Grothendieck, a towering figure in 20th-century mathematics, made significant contributions to various fields, including algebraic geometry, functional analysis, and category theory. His work has also found intriguing connections to quantum mechanics, particularly through Grothendieck's theorem and its implications.

• • Grothendieck's Theorem**: This theorem, originally formulated in the context of Banach spaces, has profound implications in quantum mechanics. It is particularly relevant in the study of Bell's inequalities, which are fundamental in understanding quantum entanglement and non-locality¹². Grothendieck's inequality helps in approximating solutions to certain problems in quantum mechanics that are otherwise computationally intractable.
  • Bell's Inequality**: In quantum mechanics, Bell's inequality tests the predictions of quantum mechanics against those of classical physics. Grothendieck's theorem provides a framework for understanding these inequalities in a more generalized mathematical context¹.
  • Non-commutative Geometry**: Grothendieck's ideas have also influenced non-commutative geometry, a field that extends classical geometry to quantum spaces. This has applications in quantum field theory and string theory, where the geometry of space-time itself can exhibit quantum properties².

Doring introduced the mathematical setting and the key physical question: what is a physically sensible and useful definition of a quantum state space?^[9]

For each quantum system described by an operator algebra A of physical quantities, we provide a (generalized) state space, notwithstanding the

Kochen–Specker theorem. This quantum state space is the spectral presheaf Σ. We formulate the time evolution of quantum systems in terms of Hamiltonian flows on this generalized space and explain how the structure of the spectral presheaf

Σ geometrically mirrors the double role played by self-adjoint operators in quantum theory, as quantum random variables and as generators of time evolution.

Source: Conversation with Copilot, 9/2/2024

(1) GROTHENDIECK’S THEOREM, PAST AND PRESENT - American Mathematical Society. https://www.ams.org/journals/bull/2012-49-02/S0273-0979-2011-01348-9/S0273-0979-2011-01348-9.pdf.
(2) [1101.4195] Grothendieck's Theorem, past and present - arXiv.org. https://arxiv.org/abs/1101.4195.

Grothendiek

The concept of the "excluded middle" is a fundamental principle in classical logic, stating that for any proposition, either that proposition is true, or its negation is true. This principle, however, does not always hold in certain mathematical frameworks and has interesting implications in both Grothendieck's work and quantum mechanics.

Grothendieck's Contributions and the Excluded Middle

Alexander Grothendieck's work, particularly in the realm of topos theory, challenges the classical logic's law of the excluded middle. Topos theory provides a framework where logic can be more flexible, allowing for the use of intuitionistic logic, which does not adhere to the law of the excluded middle. This is particularly useful in constructive mathematics, where proofs must constructively demonstrate the existence of an object rather than relying on non-constructive methods.

Applications in Quantum Mechanics

In quantum mechanics, the law of the excluded middle is often not applicable due to the nature of quantum states and superposition. Quantum mechanics operates on principles that are fundamentally different from classical logic. For instance:

1. **Superposition**: A quantum system can exist in multiple states simultaneously. This contradicts the classical notion that a system must be in one state or another (excluded middle).

2. **Quantum Logic**: The logical structure of quantum mechanics is better described by quantum logic, which modifies classical logic to accommodate the peculiarities of quantum phenomena. In quantum logic, the principle of the excluded middle does not hold in the same way as it does in classical logic².

3. **Bell's Theorem and Non-locality**: Bell's theorem and the associated experiments demonstrate that local realism (the idea that objects have definite states independent of measurement) is not compatible with quantum mechanics. This further illustrates the limitations of classical logic, including the law of the excluded middle, in describing quantum systems².

Bridging Grothendieck's Work and Quantum Mechanics

Grothendieck's approach to mathematics, particularly through topos theory, provides a more flexible logical framework that can be applied to quantum mechanics. By using intuitionistic logic, which does not rely on the law of the excluded middle, mathematicians and physicists can better model and understand quantum phenomena.

In summary, Grothendieck's contributions to mathematics, especially through topos theory, offer alternative logical frameworks that align more closely with the principles of quantum mechanics. This allows for a deeper understanding and more accurate modeling of quantum systems, which often defy classical logical principles like the excluded middle.

Would you like to explore more about Grothendieck's topos theory or specific applications in quantum mechanics?

¹: [nLab on Category Theory](https://ncatlab.org/nlab/show/applications+of+%28higher%29+category+theory) ²: [Exclusion of Excluded Middle Law and Quantum Mechanics](https://arxiv.org/pdf/2303.16076.pdf)

Source: Conversation with Copilot, 9/2/2024

(1) Exclusion of Excluded Middle Law and Quantum Mechanics. https://arxiv.org/pdf/2303.16076.pdf.
(2) applications of (higher) category theory in nLab. https://ncatlab.org/nlab/show/applications+of+%28higher%29+category+theory.
(3) IN QUANTUM LOGIC - JSTOR. https://www.jstor.org/stable/30226173.

(3) Grothendieck lines in 3d $$ \mathcal{N} $$ = 2 SQCD and the quantum K .... https://link.springer.com/article/10.1007/JHEP12%282023%29082.
(4) Grothendieck: A Short Guide to His Mathematical and ... - Springer. https://link.springer.com/referenceworkentry/10.1007/978-3-030-19071-2_27-1.
(5) undefined. https://doi.org/10.48550/arXiv.1101.4195.

References

1. ↑ Khan Academy Differentiability implies continuity https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new/ab-2-4/a/proof-differentiability-implies-continuity.
2. ↑ Anton Zeilinger, Dance of the Photons (2022-11-08) ISBN 978-0374611781
3. ↑ Daniel Castro, The Universe Is Not Locally Real, and the Physics Nobel Prize Winners Proved It Scientific American (2022-10-06) https://www.scientificamerican.com/article/the-universe-is-not-locally-real-and-the-physics-nobel-prize-winners-proved-it/
4. ↑ Dan Rockmore, How Much of the World Is It Possible to Model? (2024-01-15) The New Yorker https://www.newyorker.com/culture/annals-of-inquiry/how-much-of-the-world-is-it-possible-to-model
5. ↑ Z. Ernest, Why is entropy of the universe increasing? https://socratic.org/questions/why-is-entropy-of-universe-increasing
6. ↑ Quantum Entanglement’s New Dimension: A Topological Breakthrough https://scitechdaily.com/quantum-entanglements-new-dimension-a-topological-breakthrough/
7. ↑ The Topology of Matter Oxford https://oxfordqcmt.org/topology/
8. ↑ Stephen Wolfram, A Project to Find the Fundamental Theory of Physics (2020-06-19) ISBN‎ 978-1579550356
9. ↑ Andreas Döring, Spectral presheaves as quantum state spaces file:///C:/Users/rp_to/Downloads/d%C3%B6ring-2015-spectral-presheaves-as-quantum-state-spaces.pdf

Other Material

• See wiki page on Sensible Reality
• See wiki page on Information in Physics
• See wiki page on Complex Dimensions

Sensible Reality‎‎