Gauge Theories2.2. Reality and How to Describe ItRecapping the content of 2.1: Ancient Greeks guessed that all matter consists of tiny, identical, unbroken atoms; Dalton confirmed this idea with quantitative experiments and proposed that different elements have different atoms; Mendeleev discovered that all elements are related to each other; Thompson proved the existence of electrons; Rutherford established the existence of nucleus; A planet model of the atom was proposed that can explain the spectrum but cannot explain the stability of atoms—why a supposedly accelerating electron does not radiate energy and crashed into the nucleus. This puzzle led to the birth of quantum mechanics. We concluded 2.1 by mentioning that the order of making measurements is important, or roughly speaking, the order of doing things is important. Making love before dating and dating before making love may have different consequences as we all know. That’s what quantum mechanics all about, if you think hard enough. Therefore, quantum mechanics is not that outlandish. People talking about it but not understanding it make it sound so. Now let me show you why quantum mechanics is simple and it should be the way our nature operates.
To start, we mention that the new physics in microscopic world was constructed in the 1920s by a group of young people, many of them being German-speaking boys. Old players had become either too exhausted to follow the new ideas or too conservative to make a shift of way of thinking. Max Planck who first proposed the concept of quantum thought that his idea would be swept into the dustbin of history and decided to accept the dismaying reality that his then 20-something quantum theory would be abandoned soon. He neither made any defensive moves nor tried to find a fix. Some people suggested that Planck had lost interest in physics after 1910s because of a series of family tragedies (in 10 years from 1909 to 1919, his wife and three of four children died). Lorentz was too old to attend conferences. Although Niels Bohr has been revered by physicists for his role in the quantum revolutions, his specific contributions to this new physics are either insignificant or defective. Albert Einstein could not catch up with the tide and turned himself into a foul crier. That is the golden time of physics for the young academic proletariats and the nightmare for the establishment. New discoveries come out like avalanches, old men scream in fear and hatred but shrink away swiftly, while young boys engage head-on fights and win one battle after another. To have a feel of how great these kids were, we only need a very short list: in 1924, Louis de Broglie (32) proposed the wave-particle duality; in 1924, Wolfgang Pauli (24) proposed the existence of electron spin and exclusion principle; in 1925, Weiner Heisenberg (24), Pascual Jordan (23) and Max Born (43) proposed the correct mathematical framework of quantum mechanics called matrix mechanics; in 1926, Erwin Schrodinger (39) completed another version of quantum mechanics and solved the spectrum of hydrogen atom; in 1927, Heisenberg (26) published his famous principle of uncertainty; in 1928, Paul Dirac (26) proposed the quantum mechanics that is compatible with relativity theory, i.e., applicable to the cases where speed addition violates the ordinary rule. Dirac’s theory also solved the problem of the origin of electron spin as a consequence of relativity and led to the birth of quantum field theories. In 1930, Dirac (28) published his famous book The Principles of Quantum Mechanics, which laid the solid and concise foundation of quantum mechanics, i.e., quantum mechanics is best expressed with the functional analysis in the Hilbert space, which not only cleaned up the controversies involving Schrodinger’s wave mechanics and Heisenberg’s matrix mechanics, but also provided a consistent and straightforward tool for calculating all kinds of phenomena. In less than 10 years, the pursuit of the answer to a seemingly trivial question of why the revolving electrons in an atom do not radiate energy and crash into the nucleus brought an unprecedented revolution in physics. Let us continue the story of the last posting and explain the essence of quantum mechanics.
Ffirst allow me to murmur a little more about the order of making measurements and along the way I will explain a mysterious trick called “quantization” that has become a standard procedure in all modern physical theories, including quantum gauge theories. In classical physics, we have no trouble with the order of multiplication, e.g., nobody would doubt the legitimacy of p*x = x*p. The whole story of quantum mechanics and its ramifications, or the splendid downfall of classical physics edifice, can be condensed into a short equation: AB may not be equal to BA for two physical quantities. Of course, this is just another way of saying ‘making love before dating is different from dating before making love’. If you have trouble with this equivalence, you need to stop here and think about it for a while. When we have a result on some physical quantity, such as the position or speed or energy, we must have made a measurement on it. An unspoken premise of modern science is that only what you measure counts. One should not utter a single word on something if one has not measured it beforehand. This is the very reason why science is science, not voodoo tricks. You may say, what a big deal is it, “you say nothing about making love with that hot girl if you have not been laid yet”, isn’t it obvious? Yeah, it IS obvious. It IS dumbly obvious. However, it is the mostly unobserved rule of reasoning for the people talking about science, particularly physics. For instance, one may regard the following statement obviously and perfectly valid: “Even if I do not look at the moon, I still can assume the moon is there.” Badly wrong! Logically wrong! When you say the moon is out there, this means you have a measurement result on the moon that shows that the moon is there. Logically, one cannot make such a claim when one has not made a measurement on the moon. If you carefully examine why you believe the moon is still there even if you do not look at it is because you justify your claim based on experience rather than on logic or observation. Empirically, you assume that the moon never disappears. It is always there. However, how true is it? Imagine that we have billions of solar systems. Most of them are just the same as ours, but some of them may be in the stage of being destroyed. Now we have billions of people being asked of such a question: “Is your moon still there even if you do not see it?”, then you can safely say some of them would find their answer being proven wrong immediately—some moons are gone in their solar system because their solar systems are being destroyed. So now you should see that the statement “even if I do not look at the moon, I still can assume the moon is there” is equally absurd as “1+1 may be 3.2.” or “that girl is a great sex partner although I never slept with her.” A terser and more polite way of saying it is: “Shut up if you haven’t done the damn thing!” That is the embodiment of the true spirit of science. Do not be so confident to presume that your gal is not sleeping with other guy when she is not with you!
To extend this logic a little bit further: Reality is a collection of information that you have acquired. Nothing a priori, nothing hypothetical, nothing imagined, may come into the description on reality. This may look not surprising at all and even cliché, but people can easily go awry and cannot adhere to it consistently when confronting a specific practical problem.
What is an electron? Now you know the correct answer: an electron is a physical entity described by a number of observables. More specifically, an electron = mass + charge + spin. That’s all we know about an electron. That’s all we can assume about an electron when we talk about it. Forget about the shape, size, color, spirit. We have none of them.
Back to our dear but pranky atoms. There are electrons and nucleus in an atom. This is a fact because people performed countless experiments to validate it. The mass and charge of the electron and nucleus were measured by many people. Electrons can have different states with different energies. That is also a fact because of the atomic spectra. What else can we talk about the electrons inside an atom? Can we say that the electrons move on elliptic orbits or can we say that the electron inside a hydrogen atom has a speed of, say, 1000 km/s at this moment? No. Why? Do not forget “shut up if you haven’t made a measurement!” You may say, well, let us do a measurement on the position hence the orbit of the electron in the hydrogen atom. OK. How can we do it? If we have a super high resolution microscope, then we can look through it and find where the electron is and how it moves, right? Of course you are right. But how does a microscope operate? I know. It shines a beam of light on the object and receives the reflected light from the object you want to observe, from the reflected light we can see the object there. If you want to look at an electron, you beam light on it and see the reflected light. But do not forget that an electron is tiny and very light. When you shine a single photon on it, it is scattered away, perhaps to a place far from what it was before you beam light on it. Imagine you hit a billiard with a table tennis and make a measurement of the position of the billiard with the reflected table tennis. You know you can do it but you also know it would bring errors, sometimes disgustingly big errors.
This is the fact of life in microscopic world if you still adhere to one of the most fundamental principles in science “shut up if you haven’t done the damn thing.”
Because you have to do measurement before you dare to make a claim and because measurement may introduce uncertainty, you get different outcomes if you change the order of making two measurements, right? So the frightening statement “the order of making measurements matters” should become transparent now.
Two steps away and you are with quantum mechanics.
Step 1: How to describe this order of measurements? You are a smart gal. You say: just make a simple subtraction: see the difference between the outcomes with two different orders of measurements. Bingo! You got it. In quantum mechanics, we count the difference of the outcomes from different orders of measurements. If the outcome of measuring B before A is O_ab and the outcome of measuring A before B is O_ba, then the difference is O_ab – O_ba. That is the uncertainty caused by changing the order of measurements.
At this moment, a natural question you might ask is: Can this uncertainty specified? Or is this uncertainty measurable?
This leads to Step 2: The difference of the outcomes of the measurements of different orders is quantifiable. Heisenberg found the first such a difference. That is his famous principle of uncertainty which establishes the difference between measuring position before momentum and measuring momentum before position. His result is neat and beautiful: this difference is just the Planck’s constant (ignoring some constant). It turns out that his discovery is just one very simple, though very important, case. There are numerous physical quantities hence numerous ways of changing the orders of measurement. Each pair of physical quantities has its own uncertainty relationship. For instances, you may measure energy and time in different orders, you may measure x- and y-components of the particle position in different orders, you may measure the y- and z-components of the angular momentum of a particle in different orders etc. for each of these measurements, there exists a definite uncertainty relationship. For example, if we measure x- and y-components of the angular momentum of an electron in different orders, the uncertainty is also equal to the Planck’s constant (ignoring a constant). On the other hand, if we measure the x- and y-components of the electron position (characterized by three components x, y and z) in different orders, we get the same result. This means we may measure the two components simultaneously. Therefore, changing the order of measurement may or may not change the result. In symbolic language, you may denote the (generalized) uncertainty principle as AB – BA = Ch where h is Planck’s constant and C may or may not be zero.
If you are familiar with matrix algebra, you would immediately notice that if we assign each physical quantity with a matrix, then above uncertainty relationship becomes obvious because we all know that matrix multiplication may or may not be commutable, i.e, the commutator AB - BA may or may not be zero. Indeed, in history, it was Pascual Jordan, who realized that what Heisenberg was doing is matrix operations. Jordan insisted that all physical quantities in the microscopic world must be described by a matrix. Of course, matrix is just a dress of some more general thing, called operators. In other words, matrix is a special kind of operator.
Therefore, why we have uncertainty is because our physical quantities become operators at microscopic level. We can only obtain the ‘eigenvalues’ of these operators. Uncertainty is a consequence of the commutator of two operators. But this widespread notion is misleading. The correct way of stating it is: the uncertainty can be described with commutators. Therefore, quantum mechanics is not hard. It just quantifies a straightforward logic: doing measurements in different orders may produce different results.
Quantization is, in essence, to find the correct commutators that are related to different orders of doing measurements. Therefore, I won’t waste more spacetime on quantization because it is a profoundly simple procedure of counting the difference of doing measurements in different orders. It is not ‘the deepest mystery in science’ as you are frequently terrorized by crackpots. Quantization is nothing complicated although its significance is deep and far-reaching. We, frankly speaking, are still doing something not that different than what the physicists were doing 80 years ago.
Echoing the conclusion of 2.1, quantization is “think about the difference between making love before dating and dating before making love!”
Does that make human affection an operator?
Also try to figure out how these commutators relate to our very first question "why the accelarating electrons do not lose energy by radiation and fall into the nucleus?"