# Poynting's vector - Observable - Force?

1. Apr 24, 2005

### Blackforest

Certainly an horrible way of thinking and that's why I should like to get a fundamental critic about this:

As I can read in some threads, the Poynting’s vector is of a great importance (e.g.: to know if a radiation is present or not) in a lot of works and theories; in mine too. One of the biggest limitation in all discussions around this vector is the fact that it must be zero in any inertial frame and that we mainly stay in one frame of this kind.

I also have seen that one have to make a distinction between true and not really true physical parameter; i.e., true parameter are gauge independent. Is the Poynting’s vector gauge independent in the EM theory? [My answer to this question is yes if one can find a gauge for which the Schwarz’s condition of integrability holds along the time].

If yes, then it must be possible to associate an “observable” Ŝ to this Poynting's vector S and this observable should obey the usual law concerning an evolution along the time:
d< Ŝ >/dt = (i/hbar). <[H, Ŝ]> + <partial deriv. of Ŝ/along time>
Result: even in an inertial frame where obviously < Ŝ > = 0, one gets:
0 = (i/hbar). <[H, Ŝ]> + <partial deriv. of Ŝ/along time>
As one can demonstrate that (1/c²). partial deriv. of S/along time as the same physical units than a force per unit volume, if (1/c²). partial deriv. of Ŝ/along time can be understood as the equivalent of (1/c²). partial deriv. of S/along time in the language of the observables, then, even in inertial frames, one should have a relation like:
<partial deriv. of F/per unit of volume> equivalent to - (i/hbar.c²). <[H, Ŝ]>

What do you think of that? Is it the signature for the obligatory existence of forces that only vanish if the observable associated with the Poynting’s vector commutes with the Hamiltonian H of the system under consideration?

What are these forces? Logically, it should concern any system (set, collection) of EM waves. Also, it must concern a solely wave “travelling” in one of our laboratories and this would mean that this photon should “feel” or react to this force.

If all the demonstration made here is true (I have some doubts) then a solely photon could show a deviation each time that circumstances lead to <[H, Ŝ]> not equal to 0 in the laboratory. Has someone here a concrete example what are these circumstances looking like?
Thanks

2. Apr 24, 2005

### masudr

Why must it be zero in an inertial frame?

3. Apr 24, 2005

### Blackforest

Well; I think I was refering to considerations concerning the so-called famous ZPF. Could be that it is not always true. In fact the Poynting's vector is parallel to the speed vector of the wave and the wave has a constant speed (c) in an inertial frame.

4. Apr 24, 2005

### dextercioby

It can't be zero.It's a part of $\Theta_{\mu\nu}$ for the classical em field...Unless the field is static (electro/magneto)...

Daniel.

5. Apr 24, 2005

### Blackforest

Sorry for this question but what does $\Theta_{\mu\nu}$ mean for you? The EM strength tensor in the classical approach? If one consider the eigen(self) fluctuations of the static EM field, is the Poynting vector not zero? Despite of this, is the rest of my demonstration valid?

6. Apr 24, 2005

### pmb_phy

If the energy-momentum tensor is does not vanish then this does not imply that the Poynting vector is non-zero. The Poynting vector can be zero in one frame and non-zero in another frame. Recall that this vector determines the momentum of radiation as well as the flow of energy.

The Poynting vector is a tricky little bugger. Ya gots to be carefull with it.

Pete

7. Apr 25, 2005

### Blackforest

Coming back to the discussion.
1°) "The Poynting vector can be zero in one frame and non-zero in another frame"; what you say means: the PV is not invariant (I interprete this relatively to the problematic of a frame transformation); it doesn't realy answer to my question about the independance relatively to the EM gauge (I refer for this problematic to formula in Tanoudji)
2°) If one consider each event in "vacuum" as an EM oscillator: is it not correct to write <S> = 0 for an observer at rest at the middle point of these oscillations?
Thanks for more precise explainations