By 1974, bosonic string theory was quickly becoming a mathematical mess, and attempts to make the theory mathematically consistent caused more trouble for the model than it had already. Playing with the math introduced four conditions that should have, by all rights, spelled the end of the early string theory:
Massless particles
Tachyons, which move faster than the speed of light Fermions, such as electrons, can’t exist 25 spatial dimensions
The cause of these problems was a reasonable constraint built into string theory. No matter what else string theory did, it needed to be consistent with existing physics — namely special relativity and quantum theory.
The Standard Model of particle physics was consistent with both theories (though it still had trouble reconciling with general relativity), so string theory also had to be consistent with both. If it violated a half century of established physics, there was no way it could be a viable theory.
Physicists eventually found ways to modify the theory to be consistent with these existing physical laws. Unfortunately, these modifications resulted in the four problematic features outlined in the bulleted list. It wasn’t just that these features were possible, but that they were now seemingly essential components of the theory.
Massless particles
One side effect of creating a consistent string theory is that it had to contain certain objects that can never be brought to rest. Because mass is a measure of an object while it’s at rest, these sorts of particles are called massless particles. This would be a major problem for string theory if the massless particles predicted didn’t really exist.
Overall, though, this wasn’t a terribly disturbing problem because scientists know for certain that at least one particle exists only in a state of motion: the photon. (The gluon, though not known for certain at the time, is also a massless particle.)
Under the Standard Model of particle physics at the time, it was believed that a particle called the neutrino might have a mass of zero. (Today we know that the neutrino’s mass is slightly higher than zero.)
There was also one other possible massless particle: the graviton. The graviton is the theoretical gauge boson that could be responsible for the force of gravity under quantum field theory.
The existence of massless particles in string theory was unfortunate, but it was a surmountable problem. String theorists needed to uncover the properties of massless particles and prove that their properties were consistent with the known universe.
Tachyons
A bigger problem than massless particles was the tachyon, a particle predicted by bosonic string theory that travels faster than the speed of light. Under a consistent bosonic string theory, the mathematical formulas demanded that tachyons exist, but the presence of tachyons in a theory represents a fundamental instability in the theory. Solutions that contain tachyons will always decay into another, lower energy solution — possibly in a never-ending cycle. For this reason, physicists don’t believe that tachyons really exist, even if a theory initially looks like it contains such particles.
Strictly speaking, Einstein’s theory of relativity doesn’t absolutely forbid an object from traveling faster than the speed of light. What it says is that it would require an infinite amount of energy for an object to accelerate to the speed of light. Therefore, in a sense, the tachyon would still be consistent with relativity, because it would always be moving faster than the speed of light (and wouldn’t ever have to accelerate to that speed).
Mathematically, when calculating a tachyon’s mass and energy using relativity, it would contain imaginary numbers. (An imaginary number is the square root of a negative number.)
This was exactly how string theory equations predicted the tachyon: They were consistent only if particles with imaginary mass existed. But what is imaginary mass? What is an imaginary energy? These physical impossibilities give rise to the problems with tachyons.
The presence of tachyons is in no way unique to bosonic string theory. For example, the Standard Model contains a certain vacuum in which the Higgs boson is actually a type of tachyon as well. In this case, the theory isn’t
inconsistent; it just means that the solution that was applied wasn’t a stable solution. It’s like trying to place a ball at the top of a hill — any slight movement will cause the ball to roll into a nearby valley. Similarly, this tachyon solution decays into a stable solution without the tachyons.
Unfortunately, in the case of bosonic string theory, there was no clear way to figure out what happened during the decay, or even if the solution ended up in a stable solution after decaying into a lower energy state.
With all of these problems, physicists don’t view these tachyons as actual particles that exist, but rather as mathematical artifacts that fall out of the theory as a sign of certain types of inherent instabilities. Any solution that contains tachyons quickly decays due to these instabilities.
Some physicists (and science fiction authors) have explored notions of how to treat tachyons as actual particles, a speculative concept that will come up briefly in topic 16. But for now, just know that tachyons were one of the things that made physicists decide, at the time, that bosonic string theory was a failure.
No electrons allowed
The real flaw in bosonic string theory was the one that it’s named after. The theory predicted only the existence of bosons, not fermions. Photons could exist, but not quarks or electrons.
Every elementary particle observed in nature has a property called a spin, which is either an integer value (-1, 0, 1, 2, and so on) or a half-integer value (->2, K, and so on). Particles with integer spins are bosons, and particles with half-integer spins are fermions. One key finding of particle physics is that all particles fall into one of these two categories.
For string theory to apply to the real world it had to include both types of particles, and the original formulation didn’t. The only particles allowed under the first model of string theory were bosons. This is why it would come to be known to physicists as the bosonic string theory.
25 space dimensions, plus 1 of time
Dimensions are the pieces of information needed to determine a precise point in space. (Dimensions are generally thought of in terms of up/down, left/ right, forward/backward.) In 1974, Claude Lovelace discovered that bosonic string theory could only be physically consistent if it were formulated in 25 spatial dimensions (topic 13 delves into the idea of the additional dimensions in more depth), but so far as anyone knows, we only have three spatial dimensions!
Relativity treats space and time as a continuum of coordinates, so this means that the universe has a total of 26 dimensions in string theory, as opposed to the four dimensions it possesses under Einstein’s special and general relativity theories.
It’s unusual that this requirement would be implicit in the theory. Einstein’s relativity has three spatial dimensions and one time dimension because those are the conditions used to create the theory. He didn’t begin working on relativity and just happen to stumble upon three spatial dimensions, but rather intentionally built it into the theory from the beginning. If he’d wanted a 2-dimensional or 5-dimensional relativity, he could have built the theory to work in those dimensions.
With bosonic string theory, the equations actually demanded a certain number of dimensions to be mathematically consistent. The theory falls apart in any other number of dimensions!
The reason for extra dimensions
The reason for these extra dimensions can be seen by analogy. Consider a long, loose spring (like a Slinky), which is flexible and elastic, similar to the strings of string theory. If you lay the spring in a straight line flat on the floor and pull it outward, waves move along the length of the spring. These are called longitudinal waves and are similar to the way sound waves move through the air.
The key thing is that these waves, or vibrations, move only back and forth along the length of the spring. In other words, they’re 1-dimensional waves.
Now imagine that the spring stays on the floor, but someone holds each end. Each person can move the ends of the spring anywhere they want, so long as it stays on the floor. They can move it left and right, or back and forth, or some combination of the two. As the ends of the spring move in this way, the waves that are generated require two dimensions to describe the motion.
Finally, imagine that each person has an end of the spring but can move it anywhere — left or right, back or forth, and up or down. The waves generated by the spring require three dimensions to explain the motion. Trying to use 2-dimensional or 1-dimensional equations to explain the motion wouldn’t make sense.
In an analogous way, bosonic string theory required 25 spatial dimensions so the symmetries of the strings could be fully consistent. (Conformal symmetry is the exact name of the type of symmetry in string theory that requires this number of dimensions.) If the physicists left out any of those dimensions, it made about as much sense as trying to analyze the 3-dimensional spring in only one dimension . . . which is to say, none at all.
Dealing with the extra dimensions
The physical conception of these extra dimensions was (and still is) the hardest part of the theory to comprehend. Everyone can understand three spatial dimensions and a time dimension. Give me a latitude, longitude, altitude, and time, and I can meet you anywhere on the planet. You can measure height, width, and length, and you experience the passage of time, so you have a regular familiarity with what those dimensions represent.
What about the other 22 spatial dimensions? It was clear that these dimensions had to be hidden somehow. The Kaluza-Klein theory predicted that extra dimensions were rolled up, but rolling them up in precisely the right way to achieve results that made sense was difficult. This was achieved for string theory in the mid-1980s through the use of Calabi-Yau manifolds, as I discuss later in this topic.
No one has any direct experience with these strange other dimensions. For the idea to come out of the symmetry relationships associated with a relatively obscure new theoretical physics conjecture certainly didn’t offer much motivation for physicists to accept it. And for more than a decade, most physicists didn’t.
