The Holy Grail of physics is the unification of quantum physics and relativity, a Herculean task trying to wed together two spheres as different as night and day. On one hand we have the world of the very large. This is the world of stars, planets and galaxies-the world governed by Einstein’s relativity. On the other hand we have the world of the very small-the world of atoms, neutrons, and quarks-governed by the quantum. Each of these two realms not only describes different types of objects or different sized objects-they require different types of mathematics. Even worse-the world of stars and galaxies seems to be governed by a classical, deterministic physics which fits neatly into a beautiful geometric theory, while the world of elementary particles is governed by probability, randomness, and mysterious mathematical worlds called Hilbert spaces-the world of the quantum dice.
At first glance these two theories can hardly be thought to be describing objects that belong to one and the same universe-but they do exactly that. Stars are made of atoms that obey the laws of quantum physics. Out of chance, chaos and ghostly entanglement-the orderly structure of a galaxy somehow emerges-and if we look at the ghostly quantum particles closely-atoms and elementary particles do fall in a gravitational field. Therefore there must be a path forward to a unified new physics.
Before embarking on the path of unification, it is important to make sure that one has a complete and thorough understanding of those two pillars of physics that are already well established-quantum physics and general relativity. This understanding is necessary before moving on to explore efforts at unification such as string theory and loop quantum gravity.
Let’s take a stab at quantum theory. In my book Quantum Mechanics Demystified, I lay out the mathematical framework of quantum theory. But what is the conceptual framework-the basic building blocks that one wants to come away with before trying to put together a unified theory that describes the universe?. These are the ten keys to quantum physics. They may not be the only ones-I am simply making suggestions of key concepts. You may wish to add your own. Before we start-a brief note on notation. We will denote the state of a particle or system with a bold capital letter, such as F or G, while a scalar (a plain old number) will be denoted by an small italic letter such as a or b.
1. QUANTUM STATES CAN EXIST IN SUPERPOSITION
One principle that plays a central, absolutely vital role in quantum theory is the notion of superposition. Imagine if you will that a particle can be in two mutually exclusive states that we denote F and G. These two states could represent going through one slit or the other in the two slit experiment, or they could be two different energy states of an electron in an atom, for example. The principle of superposition tells us that the state formed by their linear combination-i.e. their sum-is also a valid quantum state. That is a system can be in the state described by H = a F + b G. A quantum superposition is a special sort of beast-when we look at the system, that is when we make a measurement, we never find it in some strange mixture of the states F and G. Rather, it is always in one state or the other. That is when the system is prepared in state H measurement will sometimes find the system in state F, while at other times, measurement will find the system in state G. Note that the numbers a and b can be complex. In key #2, we interpret the meaning of the state H and explore it further.
2. THE BORN RULE
The Born Rule tells us the probability of finding a quantum system in this state or that using a simple recipe. If a quantum system is in the state described by H = a F + b G, then the probability that the system is found in state F when a measurement is made is found by squaring a while the probability that the system is found in state G when a measurement is made is found by squaring b. It is important not to confuse the fact that the Born rule tells us how to extract probabilities from a quantum state with the notion that the state is a mere statistical mixture. A superposition state like H leads to interference effects-like the fringes seen on the screen in the double slit experiment-something a statistical mixture can’t do.
3. IDENTICAL PREPARATION OF STATES DOES NOT RESULT IN IDENTICAL MEAUREMENT RESULTS
A key concept in classical science is that if you set up an experiment in exactly the same way several times- you will get repeatable results. The probabilistic nature of quantum theory-which is inherently fundamental and is not due to a lack of precision in our measurement devices-means that in many cases, if we prepare several systems in a given state, we will get different measurement results when the experiment is run. Once again, suppose that we prepare the system in the state H = a F + b G. If we run the experiment 4 times, we might make measurements and find the results F, F, G, F. The next day if we again prepare systems in state H = a F + b G , we might instead get F, G, G, G. Now if we do the experiment a large number of times, then the relative fractions of F and G will tend to the probabilities given by squaring the coefficients a and b.
4. THE UNCERTAINTY PRINCIPLE
The Heisenberg uncertainty principle tells us that we cannot know the values of two complimentary observables with absolute precision. The quintessential example given in most textbooks is the uncertainty relation between position and momentum. In short, the uncertainty principle tells us that the more precision we use in measurement of position, the less we can know about momentum and vice versa. If we wish, we can know the value of one variable to any precision we like-but at the expense of complete uncertainty in the other variable. For example, if we choose to measure a particle’s momentum with great accuracy, then we sacrifice knowledge of the particles position. The uncertainty principle signs the death warrant of classical, deterministic physics.
5. THE SCHRODINGER EQUATION AND UNITARY EVOLUTION
Quantum states to evolve in time in a deterministic manner that is governed by the Schrodinger equation. Or in a more modern sense-the time evolution of quantum states is described by unitary evolution.
6. LEARN LINEAR ALGEBRA
The fifth key to quantum theory is that the world of quantum mechanics lies in the mathematical realm of vector spaces. This means that the mathematics of quantum physics is the mathematics of linear algebra. If you want to master quantum theory then you need to know linear algebra. Learn how to manipulate matrices, how to calculate determinants, and how to find eigenvectors and eigenvalues. Learn about abstract vector spaces. Finally, learn about special types of matrices, in particular Hermitian and Unitary matrices.
6. PHYSICAL OBSERVABLES ARE REPRESENTED BY OPERATORS
This follows on the heels of point 6. In quantum theory, physical observables like momentum and energy are represented by operators. When considering the wave function approach of the Schrodinger equation, operators are instructions to do something to a function-compute the derivative say. In the mathematically equivalent matrix mechanics derived by Heisenberg, operators are represented by matrices.
7. THE ONLY POSSIBLE RESULTS OF A MEASUREMENT ARE THE EIGENVALUES OF AN OPERATOR
Following key #7, since physical observables are represented by mathematical operators, the next logical question to ask is what are the possible measurement results as predicted by the theory? It turns out that these are the eigenvalues of the operators used to represent physical observables.
8. WHAT IS THE MEANING OF THE WAVEFUNCTION?
An important issue in quantum theory is the following. Is the wavefunction a real, physical entity? Or does it just represent our state of knowledge? The meaning of the wavefunction is an important issue to resolve before we can have a “theory of everything”.
Quantum systems interact with their environment. In doing so, the strange quantumness of their nature is lost. That is superposition and the interference that comes along with it is washed out by interactions with the environment. It is believed by many that this is how the classical world of our senses arises out of the quantum morass.
Quantum physics is filled with mysteries and perhaps the greatest mystery of all is that of entanglement-the spooky action at a distance type correlation originally put forward by Einstein and his colleagues back in 1935. If two particles A and B are entangled, then their properties become correlated. In a certain state, if you measure the spin of A and find it to be spin-up, then the spin of B is spin-down. Or if you measure A and find it to be spin-down, then B is spin-up. If that is too abstract to wrap your mind around, for a loose analogy imagine that A and B are entangled dice correlated such that they always roll the same number. Roll A and find a 3, then rolling B is guaranteed to give a 3.
Spooky action at a distance tells us that we can leave A here on earth and carry B all the way to the other side of the galaxy-and their measurement results will remain correlated. Of course this isn’t completely worked out, even assuming the particles could be protected from the environment, relativistic effects and gravity might hamper such a scenario, but spatial separation, i.e. distance alone doesn’t seem to have an effect on entanglement.
The type of correlation that results in entanglement is a bit spooky in itself. It’s not really mere correlation. Without diving into the mathematics, suffice it to say that basically the two particles loose the essence of their individual identities. In an entangled system, the system as a whole assumes an identity. It is as if the whole becomes greater than the sum of its parts. To understand the entangled system you need to understand the whole and cannot understand it from the individual components alone. This amazing phenomenon-which has been confirmed in the laboratory-is of central importance in quantum computation and quantum cryptography.
Quantum physics can be mathematically daunting. Before you go on to tackle string theory and quantum gravity-make sure you have these central ideas down.
ABOUT THE AUTHOR:
David McMahon is a physicist consulting with Sandia National Labs. He has written books on quantum mechanics, relativity, and mathematics including “Quantum Mechanics Demystified “ and “Relativity Demystified” (more info at http://www.davidmcmahonbooks.com/work2.htm)