Monday, October 02, 2006

Another Day Of Getting Older..

Lucky Ali's in his album "Aks" start off with a croony : "Yeh, sawera chale, jaise milne aa gaye mujhse sab dost mere ." Occasionally I wondered how would I react if such a thing happened. I came close to it when most of my friends either called up or wrote emails on this particular day. The last known such constellation of "dosts" was for my graduation when we had a rocking time after which slowly one after another returned home to India. The official count of people who didnt return in my orbital is 2 (including myself). We all could trade a little growth for happiness is the lesson that I learnt from them.

Dad took me by surprise when he found an old manuscript where I was taking some survey notes on Casimir Effect. It was a nice gesture from him taking the time to exactly reproduce it (including my typos/spelling mistakes). Since he sent it to me electronically I wanna store it somewhere like this blog. Its a lot of work considering all those subscript/superscript/greek symbol stuff.Here it is


Most people think vacuum is place where there is absolutely no matter, a place that contains absolutely nothing. However such a thing is impossible. So a vacuum is actually where there is very little matter . A good vcuum means almost lack of air, dust and other samples of matter. Note the keyword here almost. But one theory which caught up in 1970s was the Universe can be considered as a vacuum fluctuation. It states that the Universe and everything in it may be no more or no less than than one of those vacum fluctuation that allow collections of particles to burt forth out of nothing fo a while and then be reabsorbed into the vacuum. This idea closely comes on the heels of the idea that the Universe may be a closed system. A Universe that is born in the fireball of Big Bang,expands for a time and that contracts back into fireball and disappear is a vcuum fluctuation on a grand scale.

We know that the gravitaional energy of the Universe must be negative (i.e becase it is bound). So this negative gravitational energy must precisely cancel out the positive mass energy of all matter in it.(E=Mc²)). Hence a closed Universe has a Zero energy overall. This idea can be traced out to 19th century Physicist Ludwig Boltzmann,one of the founders of statistical mechanics. He guessed that the Universe ought to be Thermodynamical Equilibrium,(which it is not when created.) So its present form is temporary stage on the way to reach it which is allowed by statistical rules provided that ht e equilibrium is maintained on average in the long run. In 1971 Edward Tryon of city Univarsity of New York submitted a paper in Nature (vol.246,P396,1973)where he developed an Idea that Big Bang as a vacuum fluctuation,but he pointed out there the time allowed for its existance in line with delta(e) delta(t) = h/2*pi can be a very long time indeed.This concept finally led to baby universes "inflation"& concepts such as these. from "70s we re used to this idea. But this notion is not based on any definite model of what a quntized field system is and hence the present approach which is based on Hamilton-Jacobi formalism leads to very different picture. V.field: In this formalism the vacuum is a non co-variant concept.i.e It will be observed that the field is static.(i.e. this is valid only in a preferred Lorenz frame.)But no experiment abiding by the quantum theory can find a frame in which vacuum field appears static. Though unobservable,the concept of vacuum is helpful in understanding the physics of Quantum field theory. It is similar to the ether concept in physics in which Einistien pointed out that ether is compatible with special relativity if we deprive it of any mechanical qualities.When quantum mechanics is applied to a radiation field by Born,Heisenberg&Jordan the out come was

e(ns) =(ns + ½)h עs

where ns=0,1,2 gives the spectrum of energy level.

Increase of ns corresponds to emmission(or creation) of photons while decrease represents vice versa Thus the state of the radiation field can thus be specified by writing down in an ordered sequennce the number of photons present corresponding to the various frequncies.This is the "occupation number". The word 'vacuum' is a misnomer in this context since space is filled with a field which carrys energy. Now even "Grand space" is not proper as since we will see later that by a change of boundry conditions in reg.egns we can attain a state of lower energy. So even for an occupation number "Zero' the field does have a Zero point energy as given by the

∑ ' (½)hעs.

A field is a useful concept used to describe particles. The moment we talk about a field, we must be able to specify its amplitude as a function of space and time. The Amplitude could refer to a scalar field,vectorfield or spinor field. At this point we must note that thehre are two types vacuum one has to deal with:

vacuum associated with electron field

vacuum associaated with electromagnetic radiation.

In case of the electron field we deal with a spinor field ψ (sanpace,time) and in case of EM file we deal with vector field amplitude A(Spae,time).But having a field amplitude Ψ does not suffice to deal with a many paarticle system as Dirac used it to only for one electron system.To customise it for many elctron environment one must first declare Ψ as an operator and then specify its anti communator ψ (r',t').

Vaccum associated with EM field : The quantum theory as noted earlier does not
consider vaacuum s a state of emptiness.In this formalisation the vacuum state of electromagnetic field. corrresponds to all the radiation oscillators being in their quiscent state. Their occupation number is Zero and are called Zeropoint oscillator We can consider the vacuum state of field by ψ 0 us. Now on a practical level, one creates or annihilates photons in a paraticular state,say k. For this purpose one defines the annihilation operators a+k ak in terms of A. Similarly we define creation and annihilation operatoars b+k, bk in terms of ψ In this Hamilton -Jacobi formalism, an asumption of classical field is made.The assumption is that " At each instant 't' in a field has a well-defined value for all X Whatever be the state of the field.".Hence in this formalism made creation & annihilation operators araae respectively given by
ak+ = (2k)-½ (kq*k - δ/δqk )

ak- = (2k)-½ (kqk - δ/δqk* )

where k = ‌│k│ and satisfy the commonation relations:

[ ak, ak '] = ( ak+ , ak"+ )=0; (ak, ak+ ) =δkk'

In some literature the factor of K is on the right hand side where as in some it has a factor of k-1 on the left hand side. The vacuum state of the field ψ 0 is the stationary state defined by the requirement ak ψ 0 =0 for each K. where ak is the annihilation
operator.Using 1,above, we have:((kq*k+ δ/δqk* ) ψ0 = 0 for each K

or equivalency[ (─‌‌‌►2 ½ ψ (x) +δ/δ(x)] ψ 0=0 for each x

where (─‌‌‌►2 ½ ψ = v - ½ Σ ',kqk e'k .x is the limit of continuous K.

Solution of these differntial equations yields usψ 0 (q,q*,t) =N.Π k/2 e-k q k* q k ─i
or can be simplified asψ 0 (q,q*,t) =N e- ½ ∫ ψ (x) (─‌‌‌►2 )½ ψ. ψ (x).δ3x─1 Єot ---6 where the normalisation constant isN = .Π k/2 (k/Π) ½Now in these coordinates the Schroedinger's equation becomes:i ∂ψ/∂t =∑,' [ ─∂²/∂ q k ∂ k* +k2 ∂ k* q k )------------------------7 k/2 Now the energy could be found directly by direct substitution in to 7 above. to be asЄ0 = ∑,' k/2clearly this is infinte.Confirmation:The equation 5 above is a product of the modes, it is the ground state of an infinite set of independent operators.As we know that the oscillator is at rest,to confirm this,we note to a constant the phase of the wave function given by S= Є0 t.Thus the time evolution obtained as a solution of a guidance formula in Hamilton -Jaacobi formulation.

∂ψ(x,t)/\ ∂t =δ S[(ψ (x),t]/δ ψ (x)

ψ (x)= [(ψ (x),t]/ will become ∂ψ/ ∂t= δ S/δψ=0 or

ψ (x,t) = ψ (x) and hence the field is static. Thus the "Zeropoint" energy,─

δ S/ ∂t =0. This Є0 resides in a quantum +class.por.energy

Vacuum Associated with an Electron field : In the dirac theory all the -ve energy states are filled leading to the so-callead Dirac Sea. Now suppose that tahe elctron has an electrostatic energy Є defined by a function of the distance 'r'. i.e ε(r). If we now consisder a cell of unit volume at a distance r the energy content of the cell is ⅛Π ε² (r). (it is proved before). Hence the total Electrostatic energy is given by

E= ⅛Π ∫ε²(r).dv.

Now if we consider point charges i.e radius of the electron → 0 then E→∞ ;

if a=0 then Є=∞

But as the celebrated scientist Mr.Richard.P.Feynman said that the infinite energy may be annoying,but what really matters is whether there are any observable physical effects. If the changes on energy are Infinite then we are in trouble.

2 comments:

Perspective Inc. said...

wow...pretty interesting post.. :)

Paddy said...

Thanks.Am glad you find it interesting. Pop also talked about the old thingy of "Asatomaa Sadgamayaa" kinds and referred to the fact it was silent about the topic of vaccum being converted into energy (taamasa haranam took care of it i guess).anyways I dig his odd sense of humor :)