String Theory

Yujing Jiang

Since the 20th century, physicists have been aiming to formulate a unified theory of the universe. Such a theory should explain all interactions (strong, weak, electromagnetic and gravitational) between all elementary particles, as well as how their masses, charges and spins fit into a coherent framework. The Standard model was established as the best-tested theory of particle physics to date after the discovery of several particles that were theoretically predicted by the model such as the top quark, Higgs boson and W, Z bosons. However, the picture is still incomplete [1] :

• The Standard model assumes there to be a pattern of fields obeying symmetries, but why must these fields exist?
• What determines the parameters of the theory that dictate the strengths of interactions? Sure, one can always measure these parameters experimentally but that is no theoretical basis.
• Most importantly, the model fails to quantize gravity. The gravitational constant G has dimensions of length squared, meaning that it must multiply a positive power of characteristic energy or negative power of characteristic length to yield dimensionless quantities like scattering amplitudes (probability for an event)*. As a result, these quantities cannot be calculated perturbatively and will blow up at high energies/short distances, rendering the theory non-predictive.

These questions eventually motivated the formulation of string theory, which is based on the idea that point particles are actually vibrating strings of different frequencies.

Despite the lack of empirical support, it became the only known way to treat the divergence of gravitational interactions. Gravity is mediated by a spin-2 boson, which is essentially a mathematical object defined by how it transforms as one switches between reference frames—as long as it exists one may use it to construct general relativity, the theory of gravitation. Combining different modes of a string naturally gives rise to such a mathematical object, thus allowing it to include gravitational interactions. Furthermore, since a string has finite length, interacting particles can no longer be infinitely close. This gets rid of the divergence.

What was originally a mathematical tool to eliminate inconsistencies became a plausible explanation of the patterns of fields observed in nature. When “creating” a string theory, it must be physical on three domains—on the string, in space-time and in the vacuum background. [2] Firstly, one assumes the string has certain symmetries to ensure fundamental physical requirements such as the nonexistence of negative probabilities and causality. Quantum mechanics requires that these symmetries still hold when event probabilities are calculated by integrating over all “versions” of the string. Secondly, the vacuum must not spontaneously emit fields. And finally, one mustn’t count mathematically equivalent strings as separate. If all properties of the string are unchanged under some coordinate transformation on the string, the resulting string is mathematically equivalent as the original.

To satisfy all three consistency conditions, one can attach labels, called Chan–Paton factors, to the ends of open strings. These labels determine the multiplicity of strings and how they interact. The resulting pattern of strings then reproduces the kind of structures observed in the Standard Model.

String theory does however, have some strange implications:

Extra dimensions: String theory requires that spacetime has ten dimensions to be self-consistent. The reason we only observe four (three spatial, one temporal) is because the rest is compactified. Think of a spaghetti tube that has negligible radius compared to its length. From afar we can effectively assume the spaghetti to be one-dimensional, but if we look closer we notice its circumference, which is the compactified dimension.

Supersymmetry: For every known particle, there exists a partner particle with spin differing by ½ and identical properties otherwise. Without such a symmetry, a string carries unphysical states that generate negative probabilities or have imaginary mass (their mass squared is negative).

No direct experimental evidence exists for extra dimensions or supersymmetry yet.

Chirality: In 1956, Chien-Shiung Wu demonstrated experimentally that weak interactions violate parity conservation. [3] Parity conservation is the statement that physical predictions of a theory don’t change as one inverts the sign of all spatial coordinates. In the experiment, Wu monitored the decay of cobalt-60 atoms that were uniformly aligned by a magnetic field. It was found that electrons were preferentially emitted in the direction opposite the cobalt’s spin. Since parity flips electron momentum but not spin**, experimental results show that different physics would result from our world under parity, specifically, that more electrons would be emitted along the cobalt’s spin–thus breaking parity conservation.

String theory includes chiral fermions (the parity-violating particles that participate in weak interactions) by compactifying six dimensions in what is called the Calabi-Yau manifold. Its geometry naturally “prefers” certain string modes over others, allowing asymmetry to arise. Additionally, the set of string vibrational modes, Chan-Paton labels and choice of compactification also provides a way to compute the charges and masses of particles.

Aside from the miraculous predictions, string theory offers insights to an open problem in theoretical physics—The Black hole information paradox:

Hawking radiation is the process when a particle-antiparticle pair is created near a black hole and one particle gets sucked in while the other is emitted. Consider a black hole forming through some process then evaporating away entirely through Hawking radiation. The radiation is independent of the formation process, meaning that information regarding it is lost. This violates a principle in both classical and quantum physics called unitarity–that the state of a system at one point in time should determine its state at any other time–leading to a paradox.

In 1996, Strominger and Vafa showed that certain black holes can be modelled as bound states of strings and D-branes (end points of open strings).[4] Information could be carried in the microstates of the configuration, which accounts for how the strings vibrate and what branes they connect to. Counting the number of microstates exactly matches the Bekenstein-Hawking formula, which is the classical entropy of a black hole. Since the string theory model yields unitary calculations, this could be a promising solution to the information paradox.

In summary, string theory offers a fascinating and unifying framework that could explain some of the deepest mysteries of the universe. By replacing point particles with tiny vibrating strings, it naturally incorporates gravity, predicts new structures like extra dimensions and supersymmetric partners, and can produce the patterns of particles and forces observed in nature. Its use of D‑branes and string microstates provides a promising path toward resolving long-standing puzzles, such as the entropy of black holes and the information paradox. While direct experimental evidence remains elusive, string theory continues to guide theoretical physics toward a more complete and coherent picture of the cosmos, hinting at a universe far richer and more intricate than we can currently observe.

References

 *We are using Lorentz-Heaviside units where c=ħ=1. This means that the dimension of mass is equivalent to that of energy and the inverse of the dimension of length. You may check this by using well-known formulas like E=mc^2.

**Spin is an angular momentum that’s defined as the cross product between some position and momentum vector. Both of these vectors change sign under parity, leaving spin unchanged.

1. Polchinski, J. (1998). String theory: Volume 1, An introduction to the bosonic string. Cambridge University Press.
2. Polchinski, J. (1998). String theory: Volume 2, Superstring theory and beyond. Cambridge University Press.
3. Wu, C. S., Ambler, E., Hayward, R. W., Hoppes, D. D., & Hudson, R. P. (1957). Experimental test of parity conservation in beta decay. Physical Review, 105(4), 1413–1415.
4. Strominger, A., & Vafa, C. (1996). Microscopic origin of the Bekenstein–Hawking entropy. Physics Letters B, 379(1–4), 99–104. https://doi.org/10.1016/0370-2693(96)00345-0