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.

Planck units - introduction

In this post we are going to take a look at one of my current favourites in Physics, Planck units. These are one of those things that often don't get mentioned in physics documentaries, but should. These things are amazing and hopefully in this post I will explain why.

Firstly, as with all things like this, Wiki has a really good page on the subject. I would advise you to take a read of this its brilliant. To sum up Wiki, Planck units are based on 5 constants of nature;

For those who like the dramatic these are sometimes called "God's Units" because they are linked at a fundamental physical level, something that is beyond current understanding. They are listed here, this is taken straight from Wikipedia.

Table 1: Fundamental physical constants

Constant	Symbol	Dimension	Value in SI units with uncertainties[6]
Speed of light in vacuum	c	L T −1	2.99792458×108 m s−1 (exact by definition of meter)
Gravitational constant	G	L3 M−1 T −2	6.67384(80)×10−11 m3 kg−1 s−2[7]
Reduced Planck constant	ħ = h/2π where h is Planck constant	L2 M T −1	1.054571726(47)×10−34 J s[8]
Coulomb constant	(4πε0)−1 where ε0 is the permittivity of free space	L3 M T −2 Q−2	8.9875517873681764×109 kg m3 s−2 C−2 (exact by definitions of ampere and meter)
Boltzmann constant	kB	L2 M T −2 Θ−1	1.3806488(13)×10−23 J/K[9]

Key: L = length, M = mass, T = time, Q = electric charge, Θ = temperature.

Each of these is associated with a fundamental physical theory as follows;

the gravitational constant - general relativity and Newtonian gravity
the reduced Planck constant - quantum mechanics
the speed of light in a vacuum - electromagnetism and special relativity
the Coulomb constant - electrostatics
the Boltzmann constant - statistical mechanics and thermodynamics

The key above mentions Length, Mass, Time, Charge & Temperature, below are the Planck equivalents. These can be derived a number of different ways, in a previous post the idea of units was introduced. The post explored how you can look at units to get a better understanding of the underlying values. Instead of using Kg, m, s it is possible to use arbitrary units, dimensions, M, L, T (mass, length, time). This idea can be used to derive the Planck units.

Consider Planck Length, the unit of length is the meter, m , the "dimension" listed above is Length. So how do we get L? One way is to take a look at the dimensions of c, G and ħ

c - L T ⁻¹
G - L³ M⁻¹ T ⁻²
ħ - L² M T ⁻¹

so multiple G and ħ . This is the same as adding the dimensions, so you have

$ G \hbar = L^3 M^{-1} T^{-2} + L^2 M^{+1} T^{-1} = L^5 T^{-3}$

Then take 

$ c^3 = L^1 T^{-1} + L^1 T^{-1} + L^1 T^{-1} = L^3 T^{-3}$

So dividing one by the other now gives

$ \frac{G \hbar}{c^3} = L^5 T^{-3} - L^3 T^{-3} = L^2  = l_p^2$

Finally giving the Planck length,

$l_p = \sqrt{\frac{\hbar G}{c^3}}$

We can do a similar thing to get Planck mass,

$ \frac{\hbar}{G} = L^2 M^{+1} T^{-1} - L^3 M^{-1} T^{-2} = L^{-1} M^2 T^1$

Then multiply by c to give 

$\frac{\hbar c}{G} = L^{-1} M^2 T^1 + L^1 T^{-1} = M^2 = m_p^2$

So Planck mass is given by

$m_p = \sqrt{\frac{\hbar c}{G}}$

This can then be repeated to find values for time, charge, temperature, area, volume, energy, force, pressure, momentum, density, current and so on. To see a more detailed list take a look at the Wiki page.

This is not the only approach to deriving the Planck units, but for me it makes it clear that we are deriving a unit, even if the final equation looks very odd.

So, OK, this is interesting, sort off, but what is the big deal? Well, the big deal is that we still don't really know what these numbers represent, if they represent anything at all. The Planck length is mind numbingly small

l_p = 1.616199 x 10^-35 m

They do raise some interesting questions though. Is space quantized in the same way as energy? is the smallest length possible actually the Planck length? Why is the Planck charge not the same as the charge on an electron? Why is the Planck mass so large compared to the mass of an electron? Who know's, probably not that many at the moment.

What is amazing is that there are a number of things you can derive when starting out with a Planck unit. In several of my coming posts I will show some of the results I have obtained from an analysis of Planck units.