Re: Is Planks constant a function of Hubbles constant

Vera

This is indeed an interesting and thought provoking question. After some deliberation my simple answer is no, plancks constant and the speed of light is not a function of the Hubble Constant as far as we know at the moment. With a better understanding of the conditions in the early Universe, its Super-Symmetric phase, we may find that the values for these and other fundamental constants are fixed to the values they are for a good reason. At the moment this era of the Universe is not very well understood. Any way, it would be bad of me to simply state no and not give my reasoning. To help other readers I will precis some of the background you obviously understand.

The planck constant is defined from the constant of proportionality from the Planck-Einstein Equation

E=n h ν (1)

Where E is the energy of a wave of wavelength ν and h is plancks constant, 6.6262x10-34 Js and n=1,2,3,4,5,6.... The constant is often present in many equations used in Quantum Mechanics. We also have DeBroglies result that a wave of wavelength &lamba; and velocity, c, has a momentum, p, given by,

p=h / λ (2)

Basically this describes wave-particle duality, the observed fact that waves act as particles and particles as waves.

finally we have Einsteins result from Special Relativity that a particle of mass m has an energy E given by

E=Mc² (3)

Planck showed that the energy E of photons of wavelength ν did not occur in a continuous spectrum but where in fact discrete, or quantised. Erwin Schroedinger used these equations and some maths to show why, this led to the Schroedinger Wave Equation, see below, and the birth of Quantum Mechanics.

(4)

Where H-tilde is a hamiltonian operator and ψ is a wave-function. Schroedingers equation shows that a particle of mass m, from can only exist in discrete energy states, or quantum states, as its solutions are Laplacian. That is to say you can have E, 2E, 3E, 4E and so on but not 1.5E. In this formaulation we start talking about Wave Functions, as given by ψ being the superposition of all possible states the particle may have. This is what led to the interpretation of a the wave-function ψ as a probability density of the wave function existing in a given state. So particles are modelled as the probabilities of them being in a given state or energy level. The wave function ψ is not the same as the wavelength λ from (2)

The ambiguity of a quantum state, as you put it, is properly called the Heisenberg Uncertainty principle. This states that the uncertainty in position (delta x) multiplied by the uncertainty in momentum (delta p)of a particle is greater than or equal to a constant. Given by the equation

delta x * delta p >= h-bar (5)

Where h-bar is plancks constant divided by 2pi. This result occurs becuase the wave-function ψ is said to be non-localised. When not observing the wave-function it does not exist in a fixed place, as you might think of an electron existing in a fixed orbit about an atom. When a measurement is made, we interfere with it, the wave-function collapses and one state is observed. But as the wave-function is only a probability of the particle being in any one state there is an inherent uncertainty in the measurement. It is not classically a fixed value.

Well thats enough of Quantum Mechanics for the moment. The Hubble Constant is a measure of the rate of expansion of the Universe. It is defined as the constant of proportionality in the equation,

v=Hr (6)

Where v is the velocity of recession of a galaxy at distance r. An exact value for the hubble constant has yet to be measured. DeVaucoleurs et al have put a value on it of around 48 Km s^-1 Mpc^-1 The value of the Hubble Constant depends solely on the density of the Universe. The more dense the Universe is the more mass and the greater the Gravitational retardation of the expansion. The hubble constant can also be used to help measure the age of the Universe and some other facts about it. Its also worth bearing in mind that the value of the hubble constant is comparable to the speed of the Earth around the sun, its not that large.

We also must introduce the concept of the co-moving coordinate. This describes how the universe expands. The Hubble constant only measures the rate of expansion. We define a distance r to an object as

r=Rr₀ (7)

Where r₀ is the distance now, r is the distance at some other time and R is a co-moving coordinate. The universes expansion is described as R increasing, not r or r₀ , the distance to the object. This leads to the equation

H &is equivalent to; (dR/dT) / R (8)

Now, the universes rate of expansion can only be measured if H exceeds the local `peculiar' velocities of the Galaxy being measured. Typically peculiar velocities are around 500 Km per second. So H can only be measured over distances exceeding a few hundred Megaparsec. To put that into perspective, the expansion of the Universe is only measurable at a distance of 10<sup>21</sup> Km or more, thats 10 with 21 0's after it. It is impossible to measure the expansion on scales smaller than this, certainly not in the at the level of sub-atomic particles. Also, the value for H and the definition of R does not include any statement about how fast `per second' the Universe increases.

After a long pre-amble we get to the gist of my argument for a flat `no' In your statement "As the universe expands ... Thus, the ambiguity in any quantum state becomes a function of the expansion of the universe" you are saying that an affect due to the energy-density of the Universe, only measurable at vast distances, alters the inequality in eq 5. Put another way you are stating that the expansion along the co-moving coordinate given by eq 8 also `stretches' the wave-function ψ in (4) and so the probability density function is changed. This can not happen as the expansion does not occur on this small a scale. Basically your statement is patently not true if I have presented my case strongly enough. The two affects are radically different, occur on scales 10³⁰ orders of magnitude apart, and come from different formalisms. In fact the challenge today is to make sense of why these two formalisms exist so apart from each other. Hubbles and Plancks constants are that, constants, they are not functions of each other.

As to your second question/statement/hypothesis/lemma or whatever, this is so wildly at variance to accepted physics I can not even begin to comment on it. I would like to do so but there are so many assumptions and mis-conceptions in this simple paragraph that it would take weeks to untangle why you have said this.

For more details on the mathematics and concepts I would refer you to "Introductory Quantum Mechanics" by French and Taylor, "The New Physics" edited by Paul Davies and the web pages of John Baez and Ned Wrights Tutorial as well as Eric's Tutorial