Gauge Theories -2.4 Fields and Quantum Fields (2)2.4 Fields and Quantum Fields (2) As usual, let us review the main points of the previous posting: fields are the most fundamental being of the reality. Vacuua are the lowest states of fields and particles are the excited states of fields. Consequently, particles and fields are the same stuff. Creation and annihilation of particles are just a graphic way of depicting energy input to and output from fields, respectively. The order of performing measurements on quantum fields is expressed with the commutation relationship between creation and annihilation operators.
At this point, you should be able to properly answer the oft-asked questions such as “what is the size and shape of an electron?” and ”how can you be sure all electrons in the universe are the same?” etc.
No let us move on.
The previous posting concluded with this question: “since quantum field theories are just applications of quantum mechanics to systems with infinite number of degrees of freedom, why don’t we simply call them quantum mechanics?”
We will give a full answer to this question in this posting. Along the way, we will talk about relativity and why it has to be accommodated in a decent quantum theory. We hope this posting will also assist the reader to have a better understanding of the concepts introduced in the previous posting. After this posting, we will be able to introduce a set of new commutation rules, complementing the one we mentioned at the end of the previous posting.
First of all, we point out a trivial fact: the term “quantum mechanics” is normally used as synonymous with “non-relativistic quantum mechanics.” This is to say, ‘quantum mechanics’ used in most settings, does not consider the effects of relativity. “That’s certainly not good enough,” I can hear your blurting. A complete quantum theory must be consistent with the special theory of relativity and must be integrated with it. Such a quantum theory has a special name—quantum field theory.
Quantum field theory fulfills the mission of unifying quantum mechanics and special theory of relativity. According to Steven Weinberg, the master among masters, quantum field theory is the only logical consequence of quantum theory integrated with special theory of relativity (plus some other commonly assumed principles in physics). Quantum mechanics has to become quantum field theory if relativity is taken seriously. You may also sense that a quantum theory incorporated with special theory of relativity somehow necessitates fields. Yes, exactly. Fields become the most fundamental building blocks of nature once quantum theory and special theory of relativity meet, mingle and mate. If you are a curious guy, I’m sure you must be utterly intrigued at this point. A lot of people get completely lost here and at the same time desperately attracted to this seemingly unlikely twist because it sounds as if relativity plays some magic acts that turn the stage (fields) into protagonists. We should not miss the opportunity to enjoy the mental euphoria offered by the great quantum physicists.
I do not need to spend much spacetime talking about relativity theory because it is well known for being profoundly trivial. It can be loosely packed into a very short sentence: the speed of light is the same for any observers. It is always c no matter in which reference frame you perform the measurement. Suppose someone is moving at, say, 0.5c and send a beam of light forward, the other guy who is at rest measures the speed of light and he will get exactly the same value c, not c + 0.5c. One cannot add a speed to c. Why does light show such an odd behavior? Textbooks tell you that experimental results in electromagnetism say so. (But I believe there exists a ‘deeper’ interpretation and I challenge you to figure it out and post your answer here. I will show my interpretation in the next posting.) You can imagine that to maintain such a bizarre behavior of light, the rule for adding up speeds, i.e., the transformation rule from one reference frame to another must be a little bit different from that you see in high school books. That is Lorentz transformation, a simple algebraic expression even Einstein, well known for his math deficiency, could manage it properly after sweating on some trials and errors. So I do not need to waste spacetime to write down that transformation here. “Light does not change speed”, that’s all relativity theory about. The consequences of this deceptively simple statement, however, are somewhat counterintuitive albeit not mindboggling. The mostly referred phenomenon is that spacetime interval, i.e. the duration of a clock’s ticking and the length of a ruler, depend on speed. Motion shortens length and dilates ticking. It is actually a rather shallow conclusion if you must keep the speed of light the same in all frames of reference. If the ruler and clock do not behave differently in different reference frames, addition of speeds has to be just like ordinary arithmetic summation. Only by imposing a specific restriction on the frame-dependent ruler and clock, can light move at a constant speed in all reference frames. For example, the light in a moving frame might look traveling a longer distance as measured in the rest frame, but the duration the light travels for, measured in the rest frame, also increases by the same factor, maintaining the speed of light the same in both frames.
If you bother to take a look at the algebraic expression of the Lorentz transformation (I break my previous promise and show it in full here: x’=ax+bt, t’= ax + ct, y’=y, z’=z) that a kindergarten kid can understand, you may readily find that is just a rotation of a ‘rod’ in a four dimensional manifold. The length of the 4-rod is a constant. Relativity is absolutivity in the four dimensional (Minkowski) spacetime, as you may often hear. If you bother to take another look at the Lorentz transformation, then you’ll find the following slogan just crap: space and time are related to each other, not separated, to form a single four dimensional entity. (Experts know that space and time are already connected in ancient Galilean relativity.)
As a sideshow, I write a few more words further debunking relativity. It is a sad phenomenon that such a slim and simple theory as relativity is so gravely mystified, by both laymen and professionals. The spirit of relativity is actually exactly the same as that of quantum mechanics, i.e., reality is reality only after it is measured. We come to terms with quantum mechanics once we get rid of the ‘classical’ wishful thinking about physical quantities such as position and speed. We inevitably arrive at the special theory of relativity once we are serious about measuring distance and time duration. It is just another demonstration of the triumph of logic and realism. With this observation, I hope you are a little more empathetic to the fathers of quantum field theories who felt so compelling that quantum mechanics and relativity theory had to be unified.
One of the most conspicuous consequences of combining quantum mechanics and relativity is that particles may be created or destroyed. How does it come?
A shortcut, widely used in popular books and formal textbooks, is that uncertainty brings about energy fluctuation which according relativity theory implies a fluctuation of mass hence particles. The shorter time one observes, the more particles are created ephemerally. Remember what happens when we try to measure the location of an electron with higher and higher precision? We need more and more energetic photons (so their wavelengths are shorter and shorter). But a photon of sufficiently high energy may transform into an electron-positron pair which in turn may create (and annihilate) other particles. Therefore, according to this picture, relativity requires creation (and destruction) of particles.
However, this picture seems not to catch the most
important characteristic of relativity, i.e., spacetime is relative and governed by the Lorentz transformation. How quantum mechanics is integrated with a relative spacetime in a quantum field theory is unclear here. In other words, where in a quantum field theory is the concept of relative spacetime hired to play a pivotal role? We would better address this issue here. As mentioned before, the only math in relativity theory is the Lorentz transformation. A quantum field theory must observe the same transformation rule. That is called Lorentz invariance. Therefore, the integration of relativity theory into a quantum field theory is realized by Lorentz invariance. Therefore, you should not wonder all quantum field theories are born to be ‘relativistic’. Some unexpected consequences follow, such as that the probability of finding a particle is not an invariant anymore. In quantum mechanics, the probability of finding some particle in the ‘universe’ (some predetermined region) is conserved and can be normalized to 1 if needed. That is another way of saying that a particle would never disappear in the ‘universe’, or, a particle cannot be created out of blue. However, it is against what happens in a real world. Particles do disappear and particles do pop up as if from nowhere. For example, a photon can be absorbed by an atom and disappears for good. An electron can be absorbed by a proton or by a positron. A pair of electron and positron may be created from a high-energy photon. What amounts a little bit counter-intuitive is that the creation and annihilation of particles are a natural consequence of relativity. The requirement of Lorentz invariance on the dynamics of quantum systems not only permits, but demands, creation and annihilation of particles. You may have been told for billions times that relativity is far from human experience, but you now try to figure out why you can read and write in night under a light bulb. ‘Old quantum theory’ might tell you that an excited atom may radiate ‘spontaneously’ so you have light. But now you should understand that it is relativity that plays a background role for you.
Therefore, we learnt that particles can be created and destroyed as a consequence of Lorentz invariance. Particles living in the four-dimensional Minkowski world are not eternal, but ephemeral. Particles can be created and destroyed anywhere and anytime. This leads to the ‘fieldization’ of particles, i.e., we'd better simply assume that fields are the most fundamental ‘stuff’ filled in the spacetime and particles appear or disappear because the underlying field is excited or de-excited. Relativity, or more specifically, the Lorentz invariance, therefore, requires that quantum mechanics grow up to quantum field theory. Wavefunctions in quantum mechanics have to be upgraded to, or, replaced by, fields. ‘Finding a particle at position x’ in quantum mechanics has to change into “excitation of field at position x’. The notion ‘a particle disappears at position x’ is illegitimate in quantum mechanics, but the notion ‘a particle is annihilated at position x’ makes perfect sense in quantum field theory (and conforms with reality for sure). The reason of introducing creation and annihilation operators becomes obvious.
Relativity also leads to another crucial and striking consequence: for every particle, there must exist an anti-particle which has exactly the same rest mass and spin but with opposite charge of the original particle. To see this, we recollect how to excite a field and create particles. A field is excited if an adequate amount of energy is added into it. A particle is created somewhere and sometime in spacetime. Quantum mechanics tells us that we have uncertainty to specify precisely ‘somewhere and sometime’. This means we do not know exactly where and when the new particle is created. In quantum mechanics, we want to compute the ‘evolution’ of a wavefunction and that can done by calculating the Schroedinger equation, the equivalent of Newton’s law in classical mechanics. In particular, we need to compute, given the value of the wavefunction at some initial time t0, what is the wavefunction at some later time t. This is an exactly the same procedure in classical physics where we are normally asked to compute the position of an object at time t given the position at x0 at time t0. In classical physics, we have trajectories, the history of the positions. In quantum mechanics, we do not have trajectories and we only have probabilities. Therefore, in quantum mechanics, the ‘evolution’ or ‘prediction’ problem changes into “finding the wavefunction at time t.” That is equivalent to “finding the correlation of wavefunction at any time t and that at an initial time t0.” Correlation function (of wavefunctions) is not only easier to compute in many cases, but also can be directly extended to quantum field theories. Actually, the same strategy can also be employed in classical physics to increase computing efficiency and in many circumstances offer invaluable insights. (It is categorically called Green function method.) In quantum field theories, therefore, the primary goal is to find the correlation functions of the creation/annihilation operators. All calculations in quantum field theories boil down to computing these correlation functions. In quantum mechanics, the correlation function may show space-like characteristic, i.e., faster-than-light propagation (‘spooky action at distance’ as Einstein bitterly called it) is allowed in quantum mechanics because quantum mechanics, as we mentioned earlier, is supposed to be non-relativistic and does not observe the constraint of relativity. Quantum field theories, however, are born to be relativistic and must not permit such space-like correlations. Since particles are created or destroyed at random positions and instants, space-like correlations are intrinsically and inevitably present in quantum field theories. On the other hand, relativity theory is by no means violable. Therefore, the ball is in the court of the quantum field theory. An acceptable quantum field theory has to get rid of this ‘spooky action at distance’to marry with relativity. This, it turns out, is not an irrational demand, but a fortuitous blessing. After some struggle, physicists came to a perfect, or ’more perfect than perfect’, solution to this conundrum. If for every particle, an anti-particle, with exactly the same mass, spin but with opposite charge, is introduced, then the annoying ‘spooky action at distance’ disappears. The space-like correlations caused by the wanton hide-n-seek games played by the particles are exactly cancelled out by the same tricks played by their respective antiparticles. Don’t you think this is a one-stone-for-two-birds solution? This kind of elegant combo show of the human’s intelligence and Nature’s beauty is rare in the history of science and is never overpraised.
Textbooks may tell you that antiparticles guarantee an important constraint on the speed of information propagation: no information can be transmitted faster than the speed of light. In formal language, the space-like propagation (a shorthand notion for faster-than-the-speed-of-light signal transmission) is zero if antiparticles and particles are on the same footing in a quantum field theory. However, it remains a debated issue whether information propagation speed is the cause of antiparticles or antiparticles are the cause of information propagation speed. Most textbooks do not bring up this question but it is an interesting and worthwhile question. Yours sincerely prefers the former but some authors prefer the later. You make your own decision and you are safe, so far, at least.
Some people may tell you that antiparticles are just ‘holes’ in the vacuum, running backwards in time, but that notion, heuristic and entertaining as it may be, can be catastrophically misleading. Besides introducing the unphysical concept of ‘negative time’, this makes an antiparticle look surreal and even imaginary, but we know that for a given field, who is given the name particle and who the name antiparticle is completely arbitrary. Indeed, the formulation of a quantum field theory dictates that particles and antiparticles are on exactly the same footing. For the reader who is familiar with Fourier analysis, antiparticles in quantum field theory pop up as naturally as the ‘negative frequency’ components appear in a ‘signal’. It is only for the sake of custom and convenience that we specify which frequency is positive and which is negative. When a field is excited, both positive and negative frequencies are necessarily created, exactly the same as when a desk is pounded or the floor is stamped, both positive and negative vibrational frequencies are equi-probably generated. Therefore, you see, nothing mysterious and nothing hard about antiparticles. They have to be there and have to be taken into account.
Before concluding, we mention that the notion of particle is not a watertight concept. For example, you may ask: can I decompose a field in a different way other than Fourier expansion? Sure, you may. And you should. That’s why you heard of pseudo particles, quasi-particles and virtual particles. For example, a virtual particle is something like a wavelet rather than a single-frequency component (ordinary particle). A wavelet may have many different frequencies and contain a decay factor (equivalent to having a finite lifetime). Indeed, we still have some open questions about ‘particles’. We will come back to this point in due time.
With the preparation of this posting, we will be able to come back to our earlier assignment—the commutators of quantum fields.