Monday, October 31, 2005

Entropy, black holes, string theory

One of the most dramatic recent results in string theory is the derivation of the Bekenstein-Hawking entropy formula for black holes obtained by counting the microscopic string states which form a black hole. Bekenstein noted that black holes obey an "area law", dM = K dA, where 'A' is the area of the event horizon and 'K' is a constant of proportionality. Since the total mass 'M' of a black hole is just its rest energy, Bekenstein realized that this is similar to the thermodynamic law for entropy, dE = T dS. Hawking later performed a semiclassical calculation to show that the temperature of a black hole is given by T = 4 k [where k is a constant called the "surface gravity"]. Therefore the entropy of a black hole should be written as S = A/4. Physicists Andrew Strominger and Cumrin Vafa, showed that this exact entropy formula can be derived microscopically (including the factor of 1/4) by counting the degeneracy of quantum states of configurations of strings and D-branes which correspond to black holes in string theory. This is compelling evidence that D-branes can provide a short distance weak coupling description of certain black holes! For example, the class of black holes studied by Strominger and Vafa are described by 5-branes, 1-branes and open strings traveling down the 1-brane all wrapped on a 5-dimensional torus, which gives an effective one dimensional object -- a black hole.
( from: http://www.sukidog.com/jpierre/strings/bholes.htm)

Some have compared this extraordinary result to the significance of Boltzman’s kinetic theory of gases. By postulating that a gas was made of microscopic molecules, he was able to derive the ideal gas formula that had already been known using macroscopic quantities such as pressure, volume and temperature. It was this landmark idea that promoted the concept of atoms and molecules. One would hope that the Strominger and Vafa calculations will do the same for the concept of strings as the basic block of matter/energy.

Thursday, October 27, 2005

Groups, rings, modules, vector spaces, Hilbert spaces

Consider a set of elements X= {a,b,c,...}and an operator *.

Group Theory requires four axioms. They are:
i) closure law: for any a and b in X, then a*b is also in X,
ii) associative law: a*(b*c) = (a*b)*c,
iii) Identity law: there is an element e such that a*e = e*a = a,
iv) Inverse law: for any a there is an element a-1 such that a * a-1= a-1*a = e. (For ordinary multiplication, the element a cannot be equal to zero).

If a group obeys the following rule: a*b = b*a then it is said to be commutative or abelian.

A ring has two operators such as * and +.
In addition to the above four axioms for each of the two operators, it has a distributive law:
a*(b+c) = a*b+a*c.

This is the essence of high school algebra.

A module A is like a set of primitive vectors. They are multiplied by a suitable ring R of scalars.

Consider a module A= {a,b,c...} and a ring R= {k,l,m...}
Definition: An R-module A is an additive commutative group together with a function that maps (k,a) into ka, subject to the following axioms:
i) k(a+b) =ka+kb
ii) (k+m)a= ka+ma
iii) (km)a=k(ma)
iv) ea=a, where e is the identity element

More explicitly, such a module is a left module because in forming ka the scalar k is written on the left of the module a. One can re-define a right module using the same procedure to define ak, with the scalar on the right.

Now if the ring is the whole field of real numbers or the complex numbers, then the module is a vector space.

Now a Hilbert space is defined as previously mentioned, with a slightly different language.(See Hilbert Space and Quantum Mechanics). Every inner product (.,.) between two vectors on a real or complex vector space H gives rise to a norm or the length of the vector, as follows:

x = sqrt (x,x)

We say that H is a Hilbert space if it is complete with respect to this norm. Completeness in this context means that any sequence of elements of the space converges to an element in the space, in the sense that the norm of differences approaches zero.

Wednesday, October 26, 2005

Hilbert Space And Quantum Mechanics

Consider the class of infinite real sequences (a1, a2...),such that the sum of all (an)2 < infinite.

Example:
i) p=(1,1,...) does not converge, that is, 1+1+1+1+...does not converge to a point.
ii) q=(1,1/2,1/4,1/8,...) converges.

Consider two points for which its infinite real sequence converges.
p=(an) and q =(bn)

Now if one defines the function d, also called a metric, as
d(p,q)= (Σ |an - bn|2)½

This space is called a Hilbert space.

Eigenvalues

We define eigenvectors and eigenvalues as follows. Let A be an n-by-n matrix of real number or complex numbers. We say that k is an eigenvalue of A with eigenvector v if v is not zero and

Av = kv, where k is a number (real or complex).

note: we have a matrix A -- often called an operator -- multiplying a vector v equals a number k times the same vector v.


The set of all k's is called the spectrum.

In quantum mechanics, the possible states of a quantum mechanical system are represented by unit vectors (called "state vectors") residing in a Hilbert space. The exact nature of this Hilbert space is dependent on the system; for example, the state space for position and momentum states is the space of square-integrable functions, while the state space for the spin of a single electron is just the product of two complex planes. Each observable ( position, momentum, energy...) is represented by a densely-defined Hermitian linear operator acting on the state space. Each eigenstate of an observable corresponds to an eigenvector of the operator, and the associated eigenvalue corresponds to the value of the observable in that eigenstate.

For example if H, called the hamiltonian, represents the linear operator of the observable quantity energy and | v > is its eigenvector. Then

iii) H | v > = E | v >, where E, the energy of that system, is a number.

In one instance, as in the case of the hydrogen atom,

iv) H = p2 /2m + V, where p is momentum, m is the mass and V is the Coulomb potential energy.

When solving that equation by substituting iv) in iii) it yields a set of constants (Ei), which will be the energy spectrum.

It turns out that in this case, Ei has discrete values. This is one feature, discreteness, that makes quantum mechanics different from classical physics. Such observables as angular momentum, energy and spin of a particle in a bound state will have discrete values.

In every day life, a body -- take a car -- can have any energy value from zero to infinity. However an electron bounded in the hydrogen atom can sustain only discrete values. Certain values are forbidden to it.

Sunday, October 09, 2005

The theory that matters

Three outstanding features of String theory:

i) The number of dimensions is inherent into the theory. If you work in classical physics or quantum mechanics, the problem you are dealing with pretty much determines in how many dimensions you will write your equation. Say you are dealing with a particle moving along a straight line. A one-dimensional equation would be sufficed. If you are dealing with a particle moving along a surface or a body rotating about an axis, a two-dimensional equation might do the trick. In other words, you are plugging into the theory how many dimensions you need for a particular problem. Not so with String theory. It tells you plain and square that you must work in ten dimensions; otherwise the equations don't make sense.

2) In the Standard model, you need to plug in twenty parameters. One example is the ratio of the mass of a muon to the mass of an electron. These twenty parameters must be fixed, usually by some lab experiments. In String Theory you need one parameter, the length of the quantum strings. It has been conjectured that it is about the Planck size, about 10 to the exponent (-33), a decimal followed by 33 zeroes. If one would blow a proton up to the size of our sun, a quantum string would be no bigger than a baseball. This is so small that most likely it will be never measured. Nevertheless, a one-parameter theory would trump on any day a twenty-parameter theory.

3) In the twentieth century two grand theories evolved: the General Theory of Relativity in which gravity is shown as the warping of the space-time continuum; and Quantum Mechanics in which the other forces are revealed as interactions that exchange particles. In particular, for the electromagnetic force, the photons are carriers of the force; for the weak nuclear force, the W's and Z bosons; and for the force between quarks, the gluons. But gravity is the only force that stands outside of this scheme. To put gravity on an equal footing with the other forces, that is, have it as a force that would exchange particles, one would need a massless boson with spin 2. It was this particular feature of String Theory - that it does produce such a particle - that made physicists think seriously about String Theory as the theory that could unify all the forces in nature and hopefully give an explanation of the twenty parameters of the Standard Model.