Energy & Time, Momentum & Position

[LarryS]

In the Energy-Momentum 4-Vector of SR, energy is associated with the time coordinate and momentum is associated with the 3 spatial coordinates. Is this association mathematically related to the energy-time and momentum-position relationships in QM uncertainty? Thanks in advance.

 

[malawi_glenn]

no, they are related to the Space-time equations in Einsteins theory of special relativity.

i.e the proper length:
$$x^\mu x_\mu = \Delta s ^2 =\sqrt{\Delta x^2 + \Delta y^2 + \Delta z^2 - c^2 \Delta t^2}$$
which is an invariant under Lorentz transformations, comes from the contravariant 4.vector:
$x^\mu = (x,y,z,t)$ together with the metric: $g^{\mu\nu}= \text{diag}(1,1,1,-1)$

Same with momentum 4-vector squared, one obtains the rest-mass:
m^2 = E^2 - p^2

 

[Phrak]

There seems to be some sort of relationship

$$\Delta p \Delta x$$

$$(E, cp_i) \rightarrow p_\mu \ , \ E/c=p_0$$

$$(t, cx_i) \rightarrow x_\mu \ , \ x_0=ct$$

Does this generalize to this?

$$\Delta E \Delta t$$

There are also the relationships

$p = \hbar k$ relating p to x as
$$E = \hbar \omega$$ relates E to t.

p:x::E:t

But these are just yet mathematical games without any physics.

 

[malawi_glenn]

There seems to be some sort of relationship, malawi.

$$\Delta p \Delta x$$

$$E, cp_i \rightarrow p_\mu \ , \ E/c=p_0$$

$$t, cx_i \rightarrow x_\mu \ , \ x_0=ct$$

Does this generalize to this?

$$\Delta E \Delta t$$

There are also the relationships

$p = \hbar k$ relating p to x as
$$E = \hbar \omega$$ relates E to t.

But these are just yet mathematical games without any physics.

Yes, there is not 'physical' connection in that sense, and mathematically the uncertainty relations comes from commutator relations. One does not say, "hey, let's take p_0 and x_0 and combine them into an uncertainty relation"..
There is no mathematical way to go from 4-vectors to uncertainty relation here, one just identify the zeroth component of the four vector of x and p, to be the same things that are involved in the uncertainty relations, this is by accident.

But on the deeper level, Energy is the generator of time evolution, and momentum is the generator of spatial translations. From this, one can go to special theory of relativity, and one can go to Quantum mechanics. Classical mechanics is the starting point for both, here lies the connection between SR and QM.

 

[atyy]

How about that both relativity and QM come from wave equations?

 

[malawi_glenn]

How about that both relativity and QM come from wave equations?

explain :-)

SR was realized by einstein from EM waves, but the theory is not really founded on it.

 

[Phrak]

After mulling this over a bit, I googled "energy-time uncertainty principle".

At the end of a Wikipedia article

http://en.wikipedia.org/wiki/Uncertainty_principle"

is something about it, including the uncertainty as to what the meaning of $\Delta T$ should be.

Another article by Baez, 2000, may be a good one, that seems to confer.

http://math.ucr.edu/home/baez/uncertainty.html"

I only scanned the both of them.

Yet I seem to recall a relationship between metastable time (delta T) of an electron in an orbital as being inversely proportional to the energy bandwidth (delta E).

 

[atyy]

explain :-)

SR was realized by einstein from EM waves, but the theory is not really founded on it.

Let's see, I was thinking something like this.

From a wave equation we get an uncertainty principle automatically.

If the wave equation is relativistic, or comes from underlying relativistic fields, then we get 4-vectors like (w,kx,ky,kz).

It's true we can think of Minkowski spacetime as fundamental. Alternatively, if we have the dynamical laws of physics (standard model) in one inertial frame, we can infer Lorentz covariance and Minkowski spacetime.

But actually thinking this way, the uncertainty principle and 4-vectors aren't really related, since we can have non-relativistic wave equations like Schrödinger's, and the uncertainty relations will still hold, without there being any 4-vectors.