As unified electromagnetism derived c, what concepts were unified to h

[jmonlive]

I suspect that Quantum Mechanics' true theory would be understood, at least partly, by what Planck's Constant unified, like how the speed of light in vacuum was derived by Maxwell's unification of electricity and magnetism as two manifestations of one fundamental force. So, if some physics-grads could give their input?

 

[ShayanJ]

The very fact that there is a theory underlying QM is still in debate. But let's assume there is such a theory for the sake of argument. Then physicists should find it and like all such quests, they should start from things they know. Its not a good idea to start from vague things that no one knows is right or wrong!

 

[bhobba]

I am not a physics grad, but am an applied math grad and have read extensively on QM.

Planks constant is simply something to convert between units.

The full explanation can be found in Chapter 3 Ballentine - QM - A Modern Development:
https://www.amazon.com/dp/9810241054/?tag=pfamazon01-20

What that chapter shows is from probabilities must be frame independent then P = MV for some constant M. Here P is the operator -i ∂ψ/∂x and V the quantum velocity operator. Of course this is the usual definition of momentum in classical physics if M was mass m. So to make this the case you define a constant h(bar) = m/M so you have -i h(bar)∂ψ/∂x = mV. Since mV by definition is momentum this leads to -i h(bar)∂ψ/∂x as the momentum operator. And since mass has different units than velocity its simply a conversion factor between them.

The relation between QM and electromagnetism lies in something called gauge invarience:
http://quantummechanics.ucsd.edu/ph130a/130_notes/node296.html

Thanks
Bill

 

[Meir Achuz]

In QM, x and p are conjugate variables in a Fourier transformation.
They are just like x and k, or omega and t, in Fourier transforms.
There is no constant like hbar for x and k because the units of k are cm^-1.
hbar originally appeared because x and p had incompatible units.
When it is recognized that they are conjugate variables, hbar can be removed.
This is the basis of 'natural units'. hbar is then just a conversion constant between
fm and Mev^-1, with 1=197 Mev-fm, just like 1=2.54 in/cm.
c can be removed when it is recognized that x and t are just different directions in 4-space.
Then both could have the same units of either cm or sec, with the conversion constant
1=299792458 cm/sec.

 

[exponent137]

In QM, x and p are conjugate variables in a Fourier transformation.
They are just like x and k, or omega and t, in Fourier transforms.
There is no constant like hbar for x and k because the units of k are cm^-1.
hbar originally appeared because x and p had incompatible units.
When it is recognized that they are conjugate variables, hbar can be removed.
This is the basis of 'natural units'. hbar is then just a conversion constant between
fm and Mev^-1, with 1=197 Mev-fm, just like 1=2.54 in/cm.
c can be removed when it is recognized that x and t are just different directions in 4-space.
Then both could have the same units of either cm or sec, with the conversion constant
1=299792458 cm/sec.

If we choose natural units, light speed has still ever its speed and principle of uncertainty still ever exists. Thus c and hbar would not disapear, only that their values are 1.
Of course, I agree that necesity of different units (kg, m, s) disapears.

 

[bhobba]

Thus c and hbar would not disapear, only that their values are 1.

They can never disappear because they convert between units. Setting them to 1 makes equations simpler - but they are still there - just hidden.

Thanks
Bill

 

[mfb]

They can never disappear because they convert between units. Setting them to 1 makes equations simpler - but they are still there - just hidden.

Thanks
Bill

They are as much "there" as the universal height to length conversion constant 1m = 1.1 yards (approximate) if you want to express all lengths in meters and all heights in yards for whatever reason. Physics works fine completely without this constant, you can express everything in meters.

In the same way, getting rid of c means expressing times in meters (or lengths in seconds). There is nothing wrong with it. It's just not convenient for our everyday life ("I'll see you in 5 billion kilometers!"), therefore we have other time units like the second. And for historical reasons, of course...

 

[exponent137]

They are as much "there" as the universal height to length conversion constant 1m = 1.1 yards (approximate) if you want to express all lengths in meters and all heights in yards for whatever reason. Physics works fine completely without this constant, you can express everything in meters.

In the same way, getting rid of c means expressing times in meters (or lengths in seconds). There is nothing wrong with it. It's just not convenient for our everyday life ("I'll see you in 5 billion kilometers!"), therefore we have other time units like the second. And for historical reasons, of course...

We are again at analysis of trialogue of Duff, Okun, Veneziano.

But you misinterpret me: I claim. If length is in light second, (and time in seconds) c still ever exists. because still ever light is moving.
Similarly, I do not claim, that we need different units for length and height. But, angle still ever exists, similarly as speed of light still ever exist, because time and length are distinct.

 

[dextercioby]

Choosing units is a matter of convenience and writing simpler looking formulas. Nothing to debate about, really.

As for what's unified within Planck's constant, well, it's a convenient parameter which fixes units in the Born-Jordan relation [x,p] = constant with dimension of action times i times unit operator on the representation space. Why is it so small ? Well, it just shows the magnitude scale of quantum effects.

 

[bhobba]

In the same way, getting rid of c means expressing times in meters (or lengths in seconds). There is nothing wrong with it. It's just not convenient for our everyday life ("I'll see you in 5 billion kilometers!"), therefore we have other time units like the second. And for historical reasons, of course...

All true.

Thanks
Bill

 

[ChrisVer]

Well it's a weird question, since it tries to bring down different things to the "same room".
the speed of light, c, was not derived by the unification of electromagnetism. Afterall this unification (in Maxwell's form) is a wave equation. The group velocity of this wave equation happens to be c in the vacuum (in real you say that in vacuum the light runs with the speed of light). This group velocity can vary within other materials at values u<c.
c is mainly derived in Special Relativity (as the upper limit of possible velocities), and is the speed you find for massless[\B] particles, whether they are photons or not. See, even gluons travel at c, although they are not described by the normal Maxwell equations, or even neutrinos (fermions) were considered until "recently" to run at c (in real now we know they must be traveling at v<c but almost c).

Now about h... it depends how you like to see it... in my opinion h is a quantity which gives the limit of hold of the quantum effects [no unification either]. This can be seen from the fact that:
[itex][x,p]=i \hbar [/itex]
which gives the intrinsic uncertainty of particles and Heisenberg's uncertainty principle. Taking the classical limit one sets [ites]\hbar \rightarrow 0 [/itex] (very small quantum effects) and the commutation relations becomes 0, thus the momentum and position are mutually measurable variables (classical mechanics), and also you get the known classical poisson brackets relation:
{x,p}= 1.

of course their values vary, depending on your choice of units.. for example if you think that [length] and [time] are measured in the same units, you can set c=1 (and see how [time] and [length] is related in the SI)... if you think that momentum is [length]^-1 (see eg [itex]p \propto \partial [/itex]) then hbar can also be set to 1 choosing the natural units.