Passage Four
Theoretical physicists use mathematics to describe certain aspects of Nature. Sir Isaac Newton was the first theoretical physicist, although in his own time his profession was called "natural philosophy".
By Newton’s era people had already used algebra and geometry to build marvelous works of architecture, including the great cathedrals of Europe, but algebra and geometry only describe things that are sitting still. In order to describe things that are moving or changing in some way, Newton invented calculus.
The most puzzling and intriguing moving things visible to humans have always been the sun, the moon, the planets and the stars we can see in the night sky. Newton’s new calculus, combined with his "Laws of Motion", made a mathematical model for the force of gravity that not only described the observed motions of planets and stars in the night sky, but also of swinging weights and flying cannonballs in England.
Today’s theoretical physicists are often working on the boundaries of known mathematics, sometimes inventing new mathematics as they need it, like Newton did with calculus.
Newton was both a theorist and an experimentalist. He spent many long hours, to the point of neglecting his health, observing the way Nature behaved so that he might describe it better. The so-called "Newton’s Laws of Motion" are not abstract laws that Nature is somehow forced to obey, but the observed behavior of Nature that is described in the language of mathematics. In Newton’s time, theory and experiment went together.
Today the functions of theory and observation are divided into two distinct communities in physics. Both experiments and theories are much more complex than back in Newton’s time. Theorists are exploring areas of Nature in mathematics that technology so far does not allow us to observe in experiments. Many of the theoretical physicists who are alive today may not live to see how the real Nature compares with her mathematical description in their work. Today’s theorists have to learn to live with ambiguity and uncertainty in their mission to describe Nature using math.
In the 18th and 19th centuries, Newton’s mathematical description of motion using calculus and his model for the gravitational force were extended very successfully to the emerging science and technology of electromagnetism. Calculus evolved into classical field theory.
Once electromagnetic fields were thoroughly described using mathematics, many physicists felt that the field was finished, that there was nothing left to describe or explain.
Then the electron was discovered, and particle physics was born. Through the mathematics of quantum mechanics and experimental observation, it was deduced that all known particles fell into one of two classes: bosons or fermions. Bosons are particles that transmit forces. Many bosons can occupy the same state at the same time. This is not true for fermions, only one fermion can occupy a given state at a given time, and this is why fermions are the particles that make up matter. This is why solids can’t pass through one another, why we can’t walk through walls—because of Pauli repulsion-the inability of fermions (matter) to share the same space the way bosons (forces) can.
While particle physics was developing with quantum mechanics, increasing observational evidence indicated that light, as electromagnetic radiation, traveled at one fixed speed (in a vacuum) in every direction, according to every observer. This discovery and the mathematics that Einstein developed to describe it and model it in his Special Theory of Relativity, when combined with the later development of quantum mechanics, gave birth to the rich subject of relativistic quantum field theory. Relativistic quantum field theory is the foundation of our present theoretical ability to describe the behavior of the subatomic particles physicists have been observing and studying in the latter half of the 20th century.
But Einstein then extended his Special Theory of Relativity to encompass Newton’s theory of gravitation, and the result, Einstein’s General Theory of Relativity, brought the mathematics called differential geometry into physics.
General relativity has had many observational successes that proved its worth as a description of Nature, but two of the predictions of this theory have staggered the public and scientific imaginations: the expanding Universe, and black holes. Both have been observed, and both encapsulate issues that, at least in the mathematics, brush up against the very nature of reality and existence.
Relativistic quantum field theory has worked very well to describe the observed behaviors and properties of elementary particles. But the theory itself only works well when gravity is so weak that it can be neglected. Particle theory only works when we pretend gravity doesn’t exist.
General relativity has yielded a wealth of insight into the Universe, the orbits of planets, the evolution of stars and galaxies, the Big Bang and recently observed black holes and gravitational lenses. However, the theory itself only works when we pretend that the Universe is purely classical and that quantum mechanics is not needed in our description of Nature.
String theory is believed to close this gap.
Originally, string theory was proposed as an explanation for the observed relationship between mass and spin for certain particles called hadrons, which include the proton and neutron. Things didn’t work out, though, and Quantum Chromodynamics eventually proved a better theory for hadrons.
But particles in string theory arise as excitations of the string, and included in the excitations of a string in string theory is a particle with zero mass and two units of spin.
If there were a good quantum theory of gravity, then the particle that would carry the gravitational force would have zero mass and two units of spin. This has been known by theoretical physicists for a long time. This theorized particle is called the graviton.
This led early string theorists to propose that string theory be applied not as a theory of hadronic particles, but as a theory of quantum gravity, the unfulfilled fantasy of theoretical physics in the particle and gravity communities for decades. But it wasn’t enough that there be a graviton predicted by string theory. One can add a graviton to quantum field theory by hand, but the calculations that are supposed to describe Nature become useless. This is because, as illustrated in the diagram above, particle interactions occur at a single point of spacetime, at zero distance between the interacting panicles. For gravitons, the mathematics behaves so badly at zero distance that the answers just don’t make sense. In string theory, the strings collide over a small but finite distance, and the answers do make sense.
This doesn’t mean that string theory is not without its deficiencies. But the zero distance behavior is such that we can combine quantum mechanics and gravity, and we can talk sensibly about a string excitation that carries the gravitational force.
This was a very great hurdle that was overcome for late 20th century physics, which is why so many young people are willing to learn the grueling complex and abstract mathematics that is necessary to-study a quantum theory of interacting strings.
What is the difference between bosons and fermions ？