Saugata's Physics log

An essay on time : Saugata Chatterjee

Saugata — Sun, 27 Mar 2011 04:56:15 +0000

Time is something we experience every day. Curiosity about the nature of time dates pre-civilization times, more than 30,00 years ago, when men drew the phases of moon and counted days. But serious thoughts about nature of time was begun only much later, in Europe, by Socrates, who plunged the idea deep into the philosophical school of Greece and taken forward by Plato and in India. where people pondered over true nature of creation. The Greco-Roman concept of time was practical, involving measurement of time. The Indians took it many more leagues further by describing time as infinite (Lord Vishnu floating eternally and creating universe out of nothingness). The concept of infinity might have also entered Vedic culture in the context of creation and the role of time, but infinity might be used from a different perspective. The endless cycles of time divided into 4 major “Yugas” clearly depicts that ancient Hindus conceived time as endless but the only way they could incorporate the notion of infinite time is to introduce cyclicity. This way they could avoid dealing with the disturbing notion of infinity. However soon they realized that the concept of infinity was inevitable and allowed for it in the creation cycle.
But still, starting from Zeno’s granular nature of time to Vedic liquidity, the true nature of time remains elusive. One thing is certain, the measurement of time requires subdivisibility while measurement of motion requires continuity. So in a way the Greeks were concerned with measuring or the practicalities of time while the Vedic culture was more interested in its esoteric properties. I would think only about howto measure time.
Time could be assigned properties which are mathematical and properties which are purely physical (mechanical). For example, the rate of change of velocity with time is a measure of force experienced. By reciprocity this means perception of time change is induced by existence of forces. As long as there are forces in nature of fundamental or mechanical in origin, time must change.
But General Relativity runs contrary to this argument. It says as long as no force acts on a unaccelerated observer it would continue to move along geodesics which are labelled (parametrized) by its own proper time. A geodesic observer is one who moves along a path of maximum proper time. From the point of view of the geometry nothing special is claimed but a naturalness condition that the manifold be allowed to have a metric or rather the spacetime be a metric space. And any path in a manifold could be parametrized but geodesics are special because it allows an invariant notion of distance (or rather a scalar product) on that manifold which results in the choice of an unique metric. Time is defined as the affine parameter labelling the path of a geodesic observer. So, here is a paradox. GR says time could very well exist in absence of force or anything. Not only that, even if nothing is happening, Time still evolves. The solution to this apparent paradox is the question : How can someone measure the change in time if he has nothing to compare to, no hands of clock changing orientation, no moon going around Earth? He can’t !! So the bottomline is that Time keeps changing but to observe that change some other associated change is necessary in a closed configuration.
This brings us to a philosophical question.Can time be perceived without fundamental forces of nature?Relativity makes Time a coordinate and says it exists on its own. But to measure Time an observer must carry a clock. What does a clock do? It changes its configuration in a cyclic manner. What is the reason for this periodic change? How is it brought about? The change could only be induced by external (gravity) or internal (flip of ammonia; EM force) forces acting on it.
So my conjecture is that Time exists in its own accord but its existence could only be perceived by the accompanied existence of natural forces or some agent which induces a change in the state of a system. Some skeptics could ask that what if someone is locked in a featureless room for days, would that mean that time is not changing? First of all this is a ill-posed situation because the person would age, feel hunger, his biological clock in his brain induce him to sleep etc. which would tell him time is progressing. But place a stone(something which cannot feel hunger etc.) in absolute space, before creation, everywhere there is total vacuum. Now, how would that stone tell that time is evolving?
Many people with a religious inclination might be willing to believe that whatever is discussed is a copy from Book of Genesis or from Vedanta, but I am not here to prove the existence of divine entities. Instead it boils down to plain facts, that forces are a necessary ingredient in the study of Time.
CP violating forces would give the famous “Arrow of time” – a direction in which direction Time must flow. In fact any irreversible phenomenon gives a direction to Time. But since all these irreversible processes have associated increase entropy, it implies that increasing entropy is a tell tale sign for increase in Time.
But we have to ask the question- why entropy always increases or why all physical processes are irreversible? The answer is simple- so that time could increase. If a broken glass could mend itself, then time might run backwards for some time and we would not even notice it.
Irreversibility is natures way of telling that Time must increase. But why is the description purely thermodynamic- the state of system goes to the maximum density of states in the phase space?
Reason might be that the forces(gauge fields) decide on measurement of time and matter fields, which take values in the state space or ensemble decide how is Time going to evolve- backwards or forwards. And since gauge fields and chiral matter fields are dual description to the same Lie algebra, it must imply that measurement of Time and its evolution are dual to each other or 2 faces of the same coin.

(to be continued ….)

Merge pdfs in one line

Saugata — Fri, 18 Mar 2011 13:27:37 +0000

First list all the pdfs with

ls -x lec*.pdf

lecture01.pdf lecture02.pdf lecture03.pdf
lecture04.pdf lecture05.pdf lecture06.pdf

now merge directly with ghostscript :
gs -q -dNOPAUSE -dBATCH -sDEVICE=pdfwrite -sOutputFile=merged.pdf lecture01.pdf lecture02.pdf lecture03.pdf lecture04.pdf lecture05.pdf lecture06.pdf

In last I had just middle-click pasted the names from each line of ls output and not the entire output at once because it contains \r \n.

GRtensor in Maple

Saugata — Fri, 11 Mar 2011 07:18:57 +0000

GRtensor in Linux uses the .mapleinit file found in the user’s home directory. Only the path to the GRlibrary and the metrics need to be specified.

libname:=`/usr/local/maple12/grtensor/lib`, libname:
grOptionMetricPath := `/usr/local/maple12/grtensor/metrics`;

Just need to change the path in libname and grOptionMetricPath

Since version 11 Maple has been using a JAVA IDE which makes the GRtensor inputs to open in a pop-up window, which is very inconvenient. The best bet is to go back to the classic worksheet mode initiated by the command

xmaple -cw

Adding the tilde in LaTex

Saugata — Fri, 25 Feb 2011 03:43:22 +0000

Adding the tilde ~ in text is a nuisance in LaTex because LaTex uses it as a space padding character. The trick I use is a tweak in the appearance of the (otherwise too wide) $\sim$ :

{\raise.17ex\hbox{$\scriptstyle\sim$}}

Other alternatives :

This was taken from the Arbitrary LateX reference … the page also provides a good comparison sheet:

When used in \texttt, I would add a \mathtt around the tilde, to make it fit the font better:

{\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}

The difference is small but noticeable.

Plot the zeros of transcendental equations in Mathematica

Saugata — Thu, 16 Sep 2010 11:05:56 +0000

A transcendental equation is defined as function f.
It is solved not by the usual Solve routine but by the new Refine routine because it includes all the different possibilities of the unknown variables the equation has.
Assuming is used to tell the kernel that the variables XX and YY are always positive. Note the way the two conditions are clubbed. With an &&.
Now, we have to set XX and YY to some value, otherwise Mathematica would not be able to Plot it.
Then to evaluate at what point the zeros of the function f occurs using the {v,f(v)} and setting v= {the solution}.
Now, we can inject this point into the Plot by the Epilog command. Epilog command is an option for graphics functions which gives a list of graphics primitives to be rendered after the main part of the graphics is rendered.
The options used for the point size / color are standard commands.
Now mathematics plots the function f. Then renders the points of its zeros directly into the plot. So, the end effect is that we see the points where the function crosses the x-axis highlighted in RED.

Riemann normal coordinates

Saugata — Mon, 02 Aug 2010 07:23:48 +0000

The Riemann normal coordinates also called RNC are special coordinates where the first derivatives of the spacetime metric vanishes. It means that given any point in a curved manifold we can construct a coordinate system which is locally flat.

See

Lecture Notes on General Relativity

Authors: Sean M. Carroll

http://arxiv.org/abs/gr-qc/9712019

on page 57

for an introduction to RNC. However he does not derive the explicit form with the correct factors. For example, the the term proportional to the Riemann has a 1/3 factor and so on. The full expansion and a rigorous method for its derivation is given in Parker and Toms on page 134.

Leonard Parker, David Toms – Quantum Field Theory in Curved Spacetime: Quantized Fields and Gravity
Publisher: Cambridge University Press | 2009-09-21 | ISBN: 0521877873

How Hamilton invented the quaternions

Saugata — Fri, 02 Apr 2010 10:35:41 +0000

This is not my post. I copied it from John Baez’ site @ http://math.ucr.edu/home/baez/dublin/index.html#hawking. This describes how Hamilton invented the quaternions.

In 1835, at the age of 30, Hamilton had discovered how to treat complex numbers as pairs of real numbers. Fascinated by the relation between complex numbers and 2-dimensional geometry, he tried for many years to invent a bigger algebra that would play a similar role in 3-dimensional geometry. In modern language, it seems he was looking for a 3-dimensional normed division algebra. His quest built to its climax in October 1843. He later wrote to his son:

Every morning in the early part of the above-cited month, on my coming down to breakfast, your (then) little brother William Edwin, and yourself, used to ask me: “Well, Papa, can you multiply triplets?” Whereto I was always obliged to reply, with a sad shake of the head: `No, I can only add and subtract them”.

The problem was that there exists no 3-dimensional normed division algebra. He really needed a 4-dimensional algebra.

Finally, on the 16th of October, 1843, while walking with his wife along the Royal Canal to a meeting of the Royal Irish Academy in Dublin, he made his momentous discovery:

That is to say, I then and there felt the galvanic circuit of thought close; and the sparks which fell from it were the fundamental equations between i,j,k; exactly such as I have used them ever since.

And in a famous act of mathematical vandalism, he carved these equations into the stone of the Brougham Bridge:

i² = j² = k² = ijk = -1

He spent the rest of his life working on quaternions.

Three phases of Standard Model and Feynman’s prediction of strong interaction

Saugata — Mon, 08 Mar 2010 12:36:59 +0000

Standard Model predicts three phases of matter -

1. Coulomb phase – Gauge(vector) bosons are massless. U(1) gauge theory or electromagnetic theory.

2. Higgs phase – Gauge(vector) bosons are massive. Electroweak theory with its massive gauge bosons (W and Z ) is the candidate.

3. Confinement phase – No quantum numbers for free quarks. Only physical states are the bound states. This phase is the strong interaction.

While reading Michael Dine’s book I came across a very interesting piece of information. Feynman predicted as early as 1963 in a private communication with Roger Dashen that in SU(3) Yang Mills theory there would be a bound state of three quarks and also a bound state of quark anti-quark pair.

Conformal Stokes theorem

Saugata — Mon, 08 Mar 2010 12:21:00 +0000

Both the conformal Stokes’ theorem and the form of delta function δ(x) in complex space are given in page 119 of di Francesco (See equation 5.33 – 5.36).

δ(x) = 1/π ∂z’ (1/z)

where z is the complex coordinate.

z = (x + i y) /2
&
z’ (z bar) = (x – i y) /2

TT OPE for bc ghosts and massless fermions

Saugata — Mon, 08 Mar 2010 12:18:23 +0000

While calculating T(z) T(w) OPE for bc ghosts and massless fermions I came across a weird result.

I could get the central charge correct and also the T(w)/(z-w)^2 term correct but the other terms which remain does not add up to T(w) as expected.

In massless fermion case:

T = :ψ ∂ ψ:

T(z) T(w) = 1/4/(z-w)^4 + T(w)/(z-w)^2 + ( 1/2 ψ ∂² ψ + ∂ ψ ∂ ψ )/(z-w)

But,
∂ T = ψ ∂² ψ + ∂ ψ ∂ ψ

So, I cannot account for the 1/2 factor in front of ψ ∂² ψ .

In bc ghost case :

T = : 2 ∂c b + c ∂b :

T(z) T(w) = 1/4/(z-w)^4 + T(w)/(z-w)^2 + ( 4 ∂²c b + 3 ∂c ∂b – c ∂²b)/(z-w)

But,
∂T = 2 ∂²c b + 3 ∂c ∂b + c ∂²b

This means I have some terms like

T(z) T(w) = 1/4/(z-w)^4 + T(w)/(z-w)^2 + ( 4 ∂²c b + 3 ∂c ∂b – c ∂²b)/(z-w)
= 1/4/(z-w)^4 + T(w)/(z-w)^2 + [∂ T + 2( ∂²c b - c ∂²b)]/(z-w)
= 1/4/(z-w)^4 + T(w)/(z-w)^2 + ∂ [ T + 2 ( ∂c b - c ∂b) ]/(z-w)

I have no idea why the term ( ∂c b – c ∂b) should be zero.

So the problem is:
1. Whether it is correct to directly differentiate expression for T(z) to get the ∂T(z) ?
2. Is there a way to discard the total derivative using EOM or Belinfante tensor ..?

I have no idea still.