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String Theory

"But if we were part of the team that confirmed string theory, we could drink for free in any bar in any college town with a university that has a strong science program!"
     -- Rajesh Koothrappali, The Big Bang Theory S02E23: The Monopolar Expedition
String Theory is a topic in high-energy theoretical particle physics which describes the interaction of one-dimensional objects. It's origins are traced back to 1968 when G.Veneziano [1] and M. Suzuki [2] were attempting to derive the S-matrix for hadronic interactions. When two hadrons interact, they can interact by means of an s-channel or t-channel interaction.

In an s-channel interaction, particles 1 and 2 combine to form a resonance (a particle composed of more fundamental particles), which then decays into particles 3 and 4. In a t-channel interaction particles 1 and 2 exchange a force-carrying particle and then continue as particles 3 and 4. These two interactions have different interpretations, but when seen from a distance particles 1 and 2 interact and particles 3 and 4 emerge; the two processes appear identical. The calculation of the S-matrix for each process is quite different, yet they produce the same scattering amplitude. For this reason, the model describing these interactions is called the Dual Resonance model.

The Dual Resonance model provided a relation between resonances and particle scattering; the S-matrix acted as a "black box" where two particles went in and two particles came out, and what happened inside (a resonance or particle exchange) was immaterial to the outcome. The Dual Resonance model was extended by Neveu, Schwarz [3, 4], and Ramond [5], to include particles with spin, and Kaku, Yu [6, 7, 8, 9] , Lovelace [10, 11] , and Alessandrini [12], calculated the multiloop amplitudes to complete the quantum theory.

The step to string theory occured when Y. Nambu [13], and T. Goto [14], realized that the Dual Resonance model was behaving as a theory of interacting one-dimensional objects. They independently developed the classical theory of strings, and the connection between the classical theory and the previously discovered first quantized theory was made by Goldstone, Goddard, Rebbi, and Thorn [15], . However, string theory quickly fell into disfavor when it became clear that the quantum theory was consistent only in a 10 or 26 dimensional spacetime. For over a decade string theory was virtually ignored until 1984 when Green and Schwarz [16], showed that the supersymmetric version of string theory was finite to all orders in perturbation theory and was free of anomalies which restrict the spacetime dimensions if the gauge group was SO(32). They also discovered that 10 dimensional supersymmetric Einstein/Yang-Mills field theory is likewise finite and anomoly free if the gauge group is SO(32) or E8xE8. Gross, Harvey, Martinec, and Rohm [17, 18] then formulated the "Heterotic string" theory, involving only closed loop strings, and showed that this theory was anomoly free and finite with the same gauge groups. The E8xE8 gauge group is important since E8 contains SU(3)xSU(2)xU(1) of the Standard Model. This would be the ideal unified field theory were we living in a 10 dimensional universe.

The solution to the dimension problem was approached along the path of a previous theory of Kaluza and Klein, where they developed a 5 dimensional theory of gravity in which the extra dimension is compactified or 'curled up'. It was discovered that the extra dimensions of Heterotic strings form Calabi-Yau manifolds. More recently it has been shown that Heterotic string theories can be constructed in 4 dimensional spacetime without a compactification process by including an internal conformal field theory. The covariant lattice construction is one particular example of this process.

There are other features in the Standard Model which string theory must reproduce in order to be considered a viable unified field theory. One example is the so called θ-vacua of QCD. In 1975, Belavin, Polyakov, Shvarts, and Tyupkin [19], found a solution to the classical QCD field equations which was self-dual, i.e. Fμ ν = *Fμ ν = 1/2 εμ ν λ ρ Fλ ρ. This type of solution is called a pseudoparticle or instanton, since quarks appear to interact with the instanton over a brief period of time. Polyakov [20], showed that the presence of an additional term in the string Lagrangian reproduces the instanton; this term is topological in nature and is the self-intersection number of the worldsheet. Mazur and Nair [21], postulated the relation of the string instanton to the QCD instanton by showing the self-intersection number is related to the QCD instanton number. Wheater [22], showed explicitly that the string instantons were one-dimensional curves as Polyakov postulated, and Robertson [23], solved the self-dual equations to produce a particular instanton solution in terms of torus knots.

The relevance of the D=4 instanton to 3+1 dimensional QCD appears in the UA(1) problem. In QCD, prior to flavor symmetry breaking, the mass matrix vanishes and the theory is invariant under chiral U(N)xU(N) transformations (where N represents the number of flavors). The result of chiral invariance is the conservation of a current, j5μ = q γμ γ5 q, called the axial current. The conserved current implies the existence of an associated Goldstone boson. This is a problem because this particular particle is not seen in nature. Breaking the UA(1) symmetry will thus not conserve the axial current and there will be no Goldstone boson. 't Hooft [24], showed that the presence of the instanton causes a violation of chirality and thus solves the UA(1) problem.

In order to further understand how the string instanton may be related to QCD, I have performed some calculations of instanton properties. By interpreting Robertson's D=4 torus knot solutions as D=3+1 solutions (by interpreting the 4-vector Xμ(σ, τ) = [X1, X2, X3, X4] as [x, y, z, t], where x, y, z are the usual spatial coordinates and t is the proper time), I have produced a set of computer generated animations showing the instanton dynamics. I have also produced animations for two types of closed string instanton solutions.

The three types of solutions are all classified by paramatrization integers p and q of complex parametric functions F(z) and G(z) where z = σ + i τ; The torus knot solutions found by Robertson are open strings; closed type B and closed type V are closed strings whose solutions were chosen for their periodicity (closed type B were devised by myself; closed type V were devised by my research advisor, Prof. Vincent Rodgers).

Solution TypeF(z)G(z)
Torus Knotszp-zq
Closed String (type B)cosh(πz) + i sinh(πz)cosh(πz) - i sinh(πz)
Closed String (type V)eπpzeπqz
The solutions then take the form Xμ(σ, τ) = [ℑ(F(z)),ℜ(F(z)),ℜ(G(z)),ℑ(G(z))] = [x, y, z, t]. We then invert the proper time t = ℑ(G(z)), expressing it in terms of the complex variable z, and then x(σ, τ), y(σ, τ), and z(σ, τ) are then expressed in terms of the proper time t. By varying t, the coordinates [x(t),y(t),z(t)] can then be calculated and the animations below are the result of this process.
Animation Control Panel
String instanton animation

The animated instanton solutions in the Animation Control Panel above have been computed for values of p and q from 1 to 9.

The following software programs were used to generate the animations and perform calculations;



Papers:


External links to information on String Theory:

Anharmonic Oscillator

In 1995 I did some work with Andreas Soemadi and Yannick Meurice to calculate the energy levels of the anharmonic oscillator to high precision. The results of the calculations we performed are shown in the table below:
nEns.d.
01.0652855095437176888570916287995
13.3068720131529135071281216846993
25.7479592688335633047335031184889
38.3526778257857547121552577346487
411.098595622633043011086458749384
513.969926197742799300973433956881
616.954794686144151337692616508879
720.043863604188461233641421107477
823.229552179939289070647087434374
926.505554752536617417469503006772

The details and results of our work appeared in a letter to the Journal of Physics A, July 1995.
As a result of these papers, I have an Erdös Number of 5. (What is an Erdös Number?)
Bob Bacus (5) coauthored withVincent G. J. Rodgers (4)hep-th/0003250
Vincent G. J. Rodgers (4)coauthored withR. Raju Viswanathan (3) MR0908046 (88h:81205)
R. Raju Viswanathan (3) coauthored withWayne M. Lawton (2) MR1678437 (99k:73077)
Wayne M. Lawton (2) coauthored withAndrzej Schinzel (1) MR0820217 (87e:11120)
Andrzej Schinzel (1) coauthored withPaul Erdös (0) MR0126410 (23 #A3706)

Bob Bacus (5) coauthored withYannick Meurice (4)MR1352136 (96d:81034)
Yannick Meurice (4)coauthored withKenichi Konishi (3)MR0944856 (89j:81094)
Kenichi Konishi (3)coauthored withHiroshi Suzuki (2) MR1401158 (97d:81048)
Hiroshi Suzuki (2) coauthored withNoga Alon (1) MR1129114 (92k:05127)
Noga Alon (1) coauthored withPaul Erdös (0)MR0818591 (87d:11015)