The nature of photons

Amendment: I have recently discovered that some of the results given here were derived by Rindler and Schwinger ages ago. Their work related to black holes, event horizons and such things. The results I give here come about from me thinking about photon creation. It is highly likely that I have derived some results and misunderstood the physical meaning. That said I can't help thinking that there may be some truth here.

Original Post:

As far as I am aware a photon is created instantly. When one is emitted from an electron in an atom, it does so in zero time. It is not there and then... it is. There is no creation period. No time when it is actually being made.

This as always bothered me a bit. It may be the case that I am actually wrong and there is already a theory out there that shows that, yep, photons do take a certain amount of time to be created. If so, I wish someone would email me and let me know. Let's work on the basis that currently there is no such theory.

Recently I have been playing with Planck units. One of these is the Planck force given by

$ F_p = m_p g_p = \frac{c^4} G $     ....(1)

$ G$ - gravitational   constant
c - speed   of   light
$ \hbar$   -   Reduced   Planck   constant

where m_p and g_p are just the Planck mass and Planck acceleration respectively,

$ g_p =\sqrt{\frac{c^7}{G \hbar}} $    ....(2)

from this we can show that

$ g_p l_p = {c^2} $    ...(3)

l_p being the Planck length. Say we start with the Planck version of energy

$ E_p = m_p {c^2} = \hbar \omega_p $    ....(4)

Also

$ E = h \nu $    ....(5)

using

$ c = \lambda \nu $    ....(6)

gives

$ E \frac \lambda {2\pi} = \hbar c $    ....(7)

but we can also show that

$ E_p l_p = \hbar c $     ...(8)

so (7) and (8) are actually equal

$ E_p l_p = E  \frac \lambda {2\pi} $    ...(9)

set

$ l =  \frac \lambda {2\pi} $    ....(10)

and re-arrange to give

$ \frac E E_p  =  \frac {l_p} l $     ....(11)

Now here comes the bit that bothers me a little, eq (1) and (4) are the Planck unit equivalent of Newtons 2nd Law and Einstein's mass energy equation. Also dividing the traditional

$ E = m c^2 $    ... (12)

by (4) gives

$ \frac E E_p  =  \frac {m} {m_p} $    ...(13)

from (11) and (13) we obtain

$ m l = m_p l_p = \frac \hbar c $     ....(14)

(aside, take a look at Compton's equation) using (3) this gives

$ m l = m_p \frac {c^2} {g_p} $    ....(15)

define a new variable such that

$ l =  \frac {c^2} {g}  $ ....(16)

put this into (15) and sub in the Planck values to give

$ m  \frac {c^2} {g} = \sqrt {\frac {\hbar c} {G} \frac {\hbar G} {c^7}} c^2 $    ...(17)

which gives

$ g = \frac {m c^3} {\hbar}  $ .... (18)

but this just

$ g \hbar = E c  $    ....(19)

The question now arises as to what the newly defined variable in (16) actually is? From the analysis of dimensions it is an acceleration and from (3) we have

$ g_p l_p = g l  $     ....(20)

We can rearrange (19) to give

$ E = \frac {\hbar} c g = \hbar \omega  $    ... (21)

giving

$ g = \omega c   $    ....(22)

From (21) is it valid to argue that energy has an acceleration? Much in the same way that energy has a mass equivalent given by (12)? If that is the case then doesn't this imply that a photon of energy E has an associated acceleration as given by (22) ? consider this though from standard equations of motion we have

$ v^2 = u^2 + 2 a s $     ...(23)

where u is initial velocity, v is final velocity, a is acceleration and s is distance. Take u=0, v=c and g from (22) this becomes

$ c^2 = 0 + 2 \omega c s   $    ...(24)

giving

$ s = \frac {c} {2\omega} $   ... (25)

sub in (6) gives

$ s = \frac {\lambda \nu} {2 . 2 \pi \nu}  $   ...(26)

so

$ s = \frac {\lambda} {4 \pi}    $ ....(27)

where s is the distance required to accelerate the energy to the speed of light. This would imply that photons are not created instantaneously, using

$  s = \frac {u+v} {2} t  $ .... (28)

with u=0 and v=c and s given by (27) we find

$  t = \frac {\lambda} { 2 \pi c} = \frac {1} {2 \pi \nu} = \frac {1} { \omega}  $   ...(29)

This value is far from zero.

There are a number of issues here though. The variable defined in (16) is fine, it is defined.  Is its use in later equations and its physical meaning correct? The maths seems fine, but is it valid to argue that it is an acceleration of energy as shown in (21)? Can we argue that the energy described in (21) is valid for a photon? If so, then it does seem that energy is accelerated.  So the value of t in (29) would be the creation time of a photon!

The distances involved here are small, the time period is also small,  the corresponding value of g is large. For example, a photon of orange light with a frequency of 500THz, from (22) and (29)

$ s = 5.99  10^{-7} m,   g = 1.5   10^{23} ms^{-2},   t = 7.95   10^{-17} s $

Is it valid to use the equations of motion described in (23) & (28)? Should there be some changes due to quantum mechanics? We are dealing with photons, which is all QM.

There are a number of other results that follow on from this, such as explaining Doppler shift for a single photon,  though I think I will leave these for a later post.

It would be nice to think that this interpretation is correct and that photons do have a creation time, but if I am honest I really don't know.