Advanced Quantum Theory, Study notes of Quantum Mechanics

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Advanced Quantum Theory

AMATH473/673, PHYS

Achim Kempf

Department of Applied Mathematics

University of Waterloo

Canada

©^ c Achim Kempf, September 2016

(Do not copy: textbook in progress)

for a theory of quantum gravity with applications in quantum cosmology. Quantum theory is still very much a work in progress and original ideas are needed as much as ever!

Note: This course is also a refresher course for beginning graduate students, as AMATH673 at the University of Waterloo. Graduate do the same homework and write the same midterm and final but write also an essay. If you are a grad student taking this course, talk with me about the topic. Here at Waterloo, there are a number of graduate courses that build on this course. For example, I normally teach every other year Quantum Field Theory for Cosmology (AMATH872/PHYS785).

Chapter 1

A brief history of quantum theory

1.1 The classical period

At the end of the 19th century, it seemed that the basic laws of nature had been found. The world appeared to be a mechanical clockwork running according to Newton’s laws of mechanics. Light appeared to be fully explained by the Faraday-Maxwell theory of electromagnetism which held that light was a wave phenomenon. In addition, heat had been understood as a form of energy. Together, these theories constituted “Classical Physics”. Classical physics was so successful that it appeared that theoretical physics was almost complete, the only task left being to add more digits of precision. And so, Max Planck’s teacher, Scholli, advised his student against a career in physics. Soon after classical physics was overthrown.

1.2 Planck and the “Ultraviolet Catastrophe”

The limits to the validity of classical physics first became apparent in measurements of the spectrum of heat radiation. It had been known that very hot objects, such as a smith’s hot iron, are emitting light. They do because matter consists of charged particles which can act like little antennas that emit and absorb electromagnetic waves. This means that also cold objects emit and absorb electromagnetic radiation. Their heat radiation is not visible because it too weak and too red for our eyes to see. Black objects are those that absorb electromagnetic radiation (of whichever frequency range under consideration) most easily and by time reversal symmetry they are therefore also the objects that emit electromagnetic radiation of that frequency range most readily. Tea in a black tea pot cools down faster than tea in a white or reflecting tea pot. Now at the time that Planck was a student, researchers were ready to apply the laws of classical physics to a precise calculation of the radiation spectrum emitted by black bodies. To everybody’s surprise the calculations, first performed by Rayleigh and Jeans, predicted far more emission of waves of short wavelengths (such as ultraviolet)

1.4. MOUNTING EVIDENCE FOR THE FUNDAMENTAL IMPORTANCE OF H 7

1.4 Mounting evidence for the fundamental impor-

tance of h

The significance of Planck’s constant was at first rather controversial. Einstein, how- ever, was prepared to take Planck’s finding at face value. In 1906, Einstein succeeded in quantitatively explaining the photoelectric effect^2. Then, he reasoned, the light’s energy packets must be of high enough energy and therefore of high enough frequency to be able to free electrons from the metal. For irrational reasons, Einstein’s expla- nation of the photoelectric effect is the only result for which he was awarded a Nobel prize. At about the same time, work by Rutherford and others had shown that atoms consist of charged particles which had to be assumed to be orbiting another. This had led to another deep crisis for classical physics: If matter consisted of charged particles that orbit another, how could matter ever be stable? When a duck swims in circles in a pond, it continually makes waves and the production of those waves costs the duck some energy. Similarly, an electron that orbits a nucleus should continually create electromagnetic waves. Just like the duck, also the electron should lose energy as it radiates off electromagnetic waves. A quick calculation showed that any orbiting electron should rather quickly lose its energy and therefore fall into the nucleus. Finally, in 1913, Bohr was able to start explaining the stability of atoms. However, to this end he too had to make a radical hypothesis involving Planck’s constant h: Bohr hypothesized that, in addition to Newton’s laws, the orbiting particles should obey a strange new equation. The new equation says that a certain quantity calculated from the particle’s motion (the so called “action”), can occur only in integer multiples of h. In this way, only certain orbits would be allowed. In particular, there would be a smallest orbit of some finite size, and this would be the explanation of the stability of atoms. Bohr’s hypothesis also helped to explain another observation which had been made, namely that atoms absorb and emit light preferably at certain discrete frequencies.

1.5 The discovery of quantum theory

Planck’s quantum hypothesis, Einstein’s light quanta hypothesis and Bohr’s new equa- tion for the hydrogen atom all contained Planck’s h in an essential way, and none of this could be explained within the laws of classical physics. Physicists, therefore, came to suspect that the laws of classical physics might have to be changed according to some overarching new principle, in which h would play a crucial role. The new physics

(^2) Under certain circumstances light can kick electrons out of a metal’s surface. Classical physics predicted that this ability depends on the brightness of the light. Einstein’s quantum physics correctly explained that it instead depends on the color of the light: Einstein’s radical idea was that light of frequency ω comes in quanta, i.e., in packets of energy ℏω

8 CHAPTER 1. A BRIEF HISTORY OF QUANTUM THEORY

would be called quantum physics. The theoretical task at hand was enormous: One would need to find a successor to Newton’s mechanics, which would be called quan- tum mechanics. And, one would need to find a successor to Faraday and Maxwell’s electromagnetism, which would be called quantum electrodynamics. The new quan- tum theory would have to reproduce all the successes of classical physics while at the same time explaining in a unified way all the quantum phenomena, from Planck’s heat radiation formula, to the stability and the absorbtion and emission spectra of atoms. The task took more than twenty years of intense experimental and theoretical re- search by numerous researchers. Finally, in 1925, it was Heisenberg who first found “quantum mechanics”, the successor to Newton’s mechanics. (At the time, Heisenberg was a 23 year old postdoctoral fellow with a Rockefeller grant at Bohr’s institute in Copenhagen). Soon after, Schr¨odinger found a seemingly simpler formulation of quan- tum mechanics which turned out to be equivalent. Shortly after, Dirac was able to fully clarify the mathematical structure of quantum mechanics, thereby revealing the deep principles that underlie quantum theory. Dirac’s textbook “Principles of Quantum Mechanics” is a key classic. The new theory of ”Quantum Mechanics”, being the successor to Newton’s mechan- ics, correctly described how objects move under the influence of electromagnetic forces. For example, it described how electrons and protons move under the influence of their mutual attraction. Thereby, quantum mechanics explained the stability of atoms and the details of their energy spectra. In fact, quantum mechanics was soon applied to explain the periodic table and the chemical bonds. What was still needed, however, was the quantum theory of those electromagnetic forces, i.e., the quantum theoretic successor to Faraday and Maxwell’s electromag- netism. Planck’s heat radiation formula was still not explained from first principles! Fortunately, the discovery of quantum mechanics had already revealed most of the deep principles that underlie quantum theory. Following those principles, Maxwell’s theory of electromagnetism was “quantized” to arrive at quantum electrodynamics so that Planck’s formula for the heat radiation spectrum could be derived. It then became clear that quantum mechanics, i.e., the quantization of classical mechanics, was merely the starting point. Somehow, quantum mechanics would have to be upgraded to become consistent with the brand new theory of relativity which Einstein had discovered! And then it would have to be covariantly combined with the quantization of electrodynamics in order to be able to describe both matter and radiation and their interactions.

1.6 Relativistic quantum mechanics

Already by around 1900, Lorentz, Einstein and others had realized that Newton’s mechanics was in fact incompatible with Faraday and Maxwell’s theory of electromag- netism, for reasons unrelated to quantum theory, thereby contributing to the crisis of

10 CHAPTER 1. A BRIEF HISTORY OF QUANTUM THEORY

predict the existence and properties of antiparticles such as the positron!

However, the fact that particles are able to create and annihilate another in col- lisions, which had clearly been observed, was beyond the power of even relativistic quantum mechanics. It was clear that a significant enlargement of the framework of quantum theory was needed.

1.7 Quantum field theory

The way forward was called “second quantization”. The starting observation was that, in quantum mechanics, the wave functions behave completely deterministically, namely according to the Schr¨odinger equation. Given the initial wave function, one can calculate its evolution with absolute certainty. It was felt that to be able to predict the evolution of something, here the wavefunction, with absolute certainty was unusual for a quantum theory. The idea of second quantization was, therefore, to apply quantum theory to quantum theory itself. To this end, the quantum mechanical wave functions were to be treated as classical fields, much like the classical electromagnetic fields. Then, the aim was to find the quantum version of those fields. Since quantum theory was to be applied to the wave functions themselves, the amplitudes of wave functions would no longer be numbers but they would be operators instead. (An operator is a linear map on an infinite dimensional vector space). As a consequence, in quantum field theory, the amplitudes of the wave functions would be subject to uncertainty relations. One should not be able to be sure of the values of the wave function, nor should one be able to be sure of the norm of the wave function. Since in quantum mechanics the normalization of the wave function to norm one means that there is exactly one particle, somewhere, i.e., one would with second quantization not necessarily be sure how many particles there are. Roughly speaking, it is in this way that the quantum fluctuations of the wave functions themselves would then account for the creation and annihilation of particles^4. The problem of finding a quantum theory for fields had of course already been en- countered when one had first tried to find the quantum theoretic successor to Faraday and Maxwell’s electrodynamics (which was consistent with special relativity from the start). As it turned out, guided by the general principles underlying quantum mechan- ics the quantum theory of the electromagnetic fields alone was not too hard to find. Following these lines, one was eventually able to write down a unifying quantum theory both of charged particles and their antiparticles, and also of their interaction through electromagnetic quanta, i.e., photons. While this theory succeeded well in describing all the interactions, including annihilation and creation processes, it did yield much more than one had bargained for. The reason was that, since now particle number was no longer conserved, the time-energy uncertainty principle made it possible for

(^4) Yes, third and higher quantization has been considered, but with no particular successes so far.

1.7. QUANTUM FIELD THEORY 11

short time intervals that energy (and therefore all kinds of particles) could be virtually “borrowed” from the vacuum. As a consequence, the new quantum field theory, called quantum electrodynam- ics, necessarily predicted that, for example, that an electron would sometimes spon- taneously borrow energy from the vacuum to emit a photon which it then usually quickly reabsorbs. During its brief existence, this so-called “virtual” photon even has a chance to split into a virtual electron-positron pair which shortly after annihilates to become the virtual photon again. In fact, the virtual electron (or the positron) during its short existence, might actually emit and quickly reabsorb a virtual photon. That photon, might briefly split into an electron positron pair, etc etc ad infinitum. Even more intriguing is that even without a real electron to start with, the vacuum alone is predicted to have virtual particles continually appearing and disappearing! In spite of all this new complexity, it turned out that the theory’s predictions for the very simplest interactions and creation-annihilation processes were in very good agree- ment with experimental results. However, the calculation of those predicted endless chain reactions of virtual processes typically yielded divergent integrals! To take those virtual processes into account should have increased the precision of predictions. In- stead, one only obtained seemingly meaningless predictions of infinite numbers. It took the combined efforts of numerous scientists, such as Feynman, Tomanaga, Weisskopf, Dyson and others, over about twenty years, to solve this problem. It turned out that those calculations that had yielded infinities did make sense after all, if one suitably recalibrated the parameters of the theory, such as the fundamental masses and charges. This process of recalibration, called renormalization, also occurs in condensed matter physics, where it is easier to understand intuitively: Consider an electron that is traveling through a crystal. It has the usual mass and charge. But if you want to influence the electron’s motion you will find that the traveling electron behaves as if it had a several times larger mass and a smaller charge. That’s because the electron slightly deforms the crystal by slightly displacing the positive and negative charges that it passes by. It is these deformations of the crystal, which travel with the electron, which make the electron behave as if it were heavier and they also shield its charge. Also, the closer we get to the electron with our measurement device, the less is its charge shielded, i.e., the more we see of the bare charge of the electron. The key lesson here is that the masses and charges that one observes in a crystal are generally not the “bare” masses and charges that the particles fundamentally possesses. The observed masses and charges even depend on how closely one looks at the electron. Now when fundamental particles travel through the vacuum, then they deform the distribution of those virtual particles that pop in and out of existence due to the time-energy uncertainty principle. Again, this makes particles behave as if they had a different mass and a different charge. The masses and charges that are observed are not the “bare” masses and charges that the particles fundamentally possess. The observed masses and charges actually depend again on how closely one looks at the particles, i.e., at what energy one observes them, say with an accelerator. In quantum

1.8. BEYOND QUANTUM FIELD THEORY? 13

that this is all there is? Should one discourage students from a career in the subject? Certainly not! In fact, the situation resembles in many ways the situation at the time Planck was a student. We have two highly successful theories - but they are inconsis- tent! As long as we consider gravity to be a fixed background for quantum theory some calculations can be performed. Hawking’s prediction of black hole radiation is of this kind. However, once we fully take into account the dynamics of general relativity, we face a problem: The predictions of infinities in quantum field theory appear to persist. In the renormalization procedure, the limit  → 0 does no longer seem to work (not for lack of trying!). This problem is very deep. Many believe that this indicates that there actually exists a finite shortest length, , in nature, much like there is a finite fastest speed. Indeed, if we put together what we know from general relativity and what we know from quantum theory, we can conclude that we cannot even in principle devise an experimental operation that would allow us to resolve distances as small as about 10 −^35 m, which is the so-called Planck scale: Consider the task of resolving some very small structure. To this end, we need to shine on it some probing particles of very short wavelength. Due to quantum theory, the shorter the wavelength, the higher is the energy uncertainty of the probing particle. According to general relativity, energy gravitates and curves space. Thus, the probing particles will randomly curve space to the extent of their energy uncertainty. Assume now that a distance of 10−^35 m or smaller is to be resolved. A short calculation shows that to this end the probing particles would have to be of such short wavelength, i.e., of such high energy uncertainty that they would significantly curve and thereby randomly disturb the region that they are meant to probe. It therefore appears that the very notion of distance loses operational meaning at distances of 10−^35 m or so. In order to describe the structure of space-time and matter at such small scales we will need a unifying theory of quantum gravity. Much effort is being put into this. In this field of research, it is widely expected that within the unified quantum gravity the- ory there will be a need for renormalization, but not for infinite renormalization. This yet-to-be found theory of quantum gravity may also solve several other major prob- lems of quantum theory. In particular, it could yield an explanation for the particular masses and charges of the elementary particles, and perhaps even an explanation for the statistical nature of quantum theoretical predictions. A very concrete major problem awaiting resolution in the theory of quantum grav- ity is the derivation of the cosmological constant, which represents the energy of the vacuum. Quantum field theory predicts the vacuum to possess significant amounts of energy due to vacuum fluctuations: Each field can be mathematically decomposed into a collection of quantum theoretical harmonic oscillators, each of which contributes a finite ground state energy of ℏω/2. General relativity predicts that the vacuum energy should gravitate, just like any other form of energy. Evidence from recent astronomical observations of the expansion rate of the universe indicates that the cosmological constant has a small but nonzero value. How much

14 CHAPTER 1. A BRIEF HISTORY OF QUANTUM THEORY

vacuum energy does quantum field theory predict? Straightforwardly, quantum field theory predicts the vacuum energy density to be infinite. If we augment quantum field theory by the assumption that the Planck length is the shortest length in nature, then quantum field theory predicts a very large vacuum energy. In fact, it is by a factor of about 10^120 larger than what is experimentally observed. This is the today’s “ultraviolet catastrophe”. It appears that whoever tries to reconcile quantum theory with general relativity must be prepared to question the very foundations of all we know of the laws of nature. Original ideas are needed that may be no less radical than those of Planck or Einstein. Current attempts are, for example, string theory and loop quantum gravity.

1.9 Experiment and theory

In the past, progress in the search for the theory of quantum gravity has been severely hampered by the fact that one cannot actually build a microscope with sufficiently strong resolving power to probe Planck scale physics. Even the best microscopes today, namely particle accelerators, can resolve distances only down to at most 10−^20 m, which is still very far from the Planck scale of 10−^35 m. Of course, guidance from experiments is not strictly necessary, as Einstein demonstrated when he first developed general relativity. Nevertheless, any candidate theory must be tested experimentally before it can be given any credence. In this context, an important recent realization was that there are possibilities for experimental access to the Planck scale other than through accelerators! One possibility could be the study of the very highly energetic cosmic rays that occasionally hit and locally briefly light up the earth’s atmosphere. Another recently much discussed possibility arises from the simple fact that the universe itself was once very small and has dramatically expanded since. The idea is, roughly speaking, that if the early expansion was rapid enough then the universe might have acted as a microscope by stretching out everything to a much larger size. Astronomical evidence obtained over the past few years indicate that this did happen. The statistical distribution of matter in the universe is currently being measured with great precision, both by direct observation of the distribution of galaxies, and through the measurement of the cosmic microwave background. Experimental evidence is mounting for the theory that the matter distribution in the universe agrees with what one would expect if it originated as tiny primordial quantum fluctuations - which were inflated to cosmic size by a very rapid initial expansion of the universe! It appears that the universe itself has acted as a giant microscope that enlarged initially small quantum phenomena into an image on our night sky! It is just possible that even the as yet unknown quantum phenomena of Planck length size have left an imprint in this image. Some of my own research is in this area. New satellite based telescopes are currently further exploring these phenomena.

16 CHAPTER 2. CLASSICAL MECHANICS IN HAMILTONIAN FORM

Heisenberg and Schr¨odinger, they did not at all look similar to classical mechanics. It was Dirac who first clarified the mathematical structure of quantum mechanics and thereby its relation to classical mechanics. Dirac remembered that a more abstract formulation of classical mechanics than Newton’s had long been developed, namely Hamiltonian’s formulation of classical mechanics. Hamilton’s formulation of classical mechanics made use of a mathematical tool called Poisson brackets. Dirac showed that the laws of classical mechanics, once formulated in their Hamiltonian form, can be upgraded by suitably introducing h into its equations, thereby yielding quantum mechanics correctly. In this way, Dirac was able to show how quantum mechanics naturally supersedes classical mechanics while reproducing the successes of classical mechanics. We will follow Dirac in this course^1.

2.2 Levels of abstraction

In order to follow Dirac’s thinking, let us consider the levels of abstraction in math- ematical physics: Ideally, one starts from abstract laws of nature and at the end one obtains concrete number predictions for measurement outcomes. In the middle, there is usually a hierarchy of mathematical problems that one has to solve. In particular, in Newton’s formulation of classical mechanics one starts by writing down the equations of motion for the system at hand. The equations of motion will generally contain terms of the type mx¨ and will therefore of the type of differential equations. We begin our calculation by solving those differential equations, to obtain functions. These functions we then solve for variables. From those variables we even- tually obtain some concrete numbers that we can compare with a measurement value. The hierarchy of abstraction is, therefore:

Differential equations ⇓ Functions ⇓ Variables ⇓ Numbers

This begs the question if there is a level of abstraction above that of differential equa- tions? Namely, can the differential equations of motion be obtained as the solution of

(^1) Actually, Schr¨odinger in his paper introducing the Schr¨odinger equation already tried to motivate his equation by an analogy with some aspect of Hamilton’s work (the so called Hamilton Jacobi theory). This argument did not hold up. In fact, he came to correctly guess his equation by following his intuition that the discreteness of quantum phenomena might mathematically arise as discrete eigenvalues - which had been known to arise in solving wave equations.

2.3. CLASSICAL MECHANICS IN HAMILTONIAN FORMULATION 17

some higher level mathematical problem? The answer is yes, as Dirac remembered: Already in the first half of the 19th century, Lagrange, Hamilton and others had found this higher level formulation of classical mechanics. Their methods had proven useful for solving the dynamics of complicated systems, and some of those methods are still being used, for example, for the calculation of satellite trajectories. Dirac thought that if Newton’s formulation of classical mechanics was not upgradable, it might be worth investigating if the higher level formulation of Hamilton might be upgradable to ob- tain quantum mechanics. Dirac succeeded and was thereby able to clearly display the similarities and differences between classical mechanics and the quantum mechanics of Heisenberg and Schr¨odinger. To see this is our first goal in this course. Remark: For completeness, I should mention that there are two equivalent ways to present classical mechanics on this higher level of abstraction: One is due to Hamilton and one is due to Lagrange. Lagrange’s formulation of classical mechanics is also upgradable, i.e., that there is a simple way to introduce h to obtain quantum mechanics from it, as Feynman first realized in the 1940s. In this way, Feynman discovered a whole new formulation of quantum mechanics, which is called the path integral formulation. I will explain Feynman’s formulation of quantum mechanics later in the course.

2.3 Classical mechanics in Hamiltonian formulation

2.3.1 The energy function H contains all information

What was Hamilton’s higher level of abstraction? How can classical mechanics be for- mulated so that Newton’s differential equations of motion are themselves the solution of a higher level mathematical problem? Hamilton’s crucial observation was the fol- lowing: the expression for the total energy of a system already contains the complete information about that system! In particular, if we know a system’s energy function, then we can derive from it the differential equations of motion of that system. In Hamilton’s formulation of classical mechanics the highest level description of a system is therefore through its energy function. The expression for the total energy of a system is also called the Hamiltonian. The hierarchy of abstraction is now:

Hamiltonians ⇓ Differential equations ⇓ Functions ⇓ Variables ⇓ Numbers

2.3. CLASSICAL MECHANICS IN HAMILTONIAN FORMULATION 19

But what is the technique with which one can derive the equations of motion from a Hamiltonian, for example, Eqs.2.1-2.3 from Eq.2.4? Exactly how does the generator, H, of the time evolution generate the time evolution equations Eqs.2.1-2.3?

2.3.2 The Poisson bracket

The general procedure by which the equations of motion can be derived from a Hamil- tonian H requires the use of a powerful mathematical operation, called “Poisson bracket”^3 : The Poisson bracket is a particular kind of multiplication: Assume that f and g are polynomials in terms of the positions and momenta of the system, say f = − 2 p 1 and g = 3x^21 + 7p^43 − 2 x^32 p^31 + 6. Then, the Poisson bracket of f and g is written as {f, g} and the evaluation of the bracket will yield another polynomial in terms of the position and momenta of the system. In this case:

{− 2 p 1 , 3 x^21 + 7p^43 − 2 x^32 p^31 + 6} = 12x 1 (2.5)

But how does one evaluate such a Poisson bracket to obtain this answer? The rules for evaluating Poisson brackets are tailor-made for mechanics. There are two sets of rules:

A) By definition, for each particle, the Poisson brackets of the positions and momenta are:

{xi, pj } = δi,j (2.6) {xi, xj } = 0 (2.7) {pi, pj } = 0 (2.8)

for all i, j ∈ { 1 , 2 , 3 }. Here, δi,j is the Kronecker delta, which is 1 if i = j and is 0 if i 6 = j. But these are only the Poisson brackets between linear terms. How to evaluate then the Poisson bracket between two polynomials? The second set of rules allow us to reduce this general case to the case of the Poisson brackets between linear terms:

B) By definition, the Poisson bracket of two arbitrary expressions in the positions and momenta, f (x, p) and g(x, p), obey the following rules:

{f, g} = − {g, f } antisymmetry (2.9) {cf, g} = c {f, g}, for any number c linearity (2.10) {f, g + h} = {f, g} + {f, h} addition rule (2.11) (^3) Remark: In elementary particle physics there is a yet higher level of abstraction, which allows one to derive Hamiltonians. The new level is that of so-called “symmetry groups”. The Poisson bracket operation plays an essential role also in the definition of symmetry groups. (Candidate quantum gravity theories such as string theory aim to derive these symmetry groups from a yet higher level of abstraction which is hoped to be the top level.)

20 CHAPTER 2. CLASSICAL MECHANICS IN HAMILTONIAN FORM

{f, gh} = {f, g}h + g{f, h} product rule (2.12) 0 = {f, {g, h}} + {h, {f, g}} + {g, {h, f }} Jacobi id. (2.13)

Let us postpone the explanation for why these definitions had to be chosen in exactly this way^4. For now, note that an immediate consequence of these rules is that the Poisson bracket of a number always vanishes:

{c, f } = 0 if c is a number (2.14)

The point of the second set of rules is that we can use them to successively break down the evaluation of a Poisson bracket like that of Eq.2.5 into sums and products of expressions that can be evaluated by using the first set of rules, Eqs.2.6,2.7,2.8. Using the product rule we immediately obtain, for example:

{x 3 , p^23 } = {x 3 , p 3 }p 3 + p 3 {x 3 , p 3 } = 1p 3 + p 3 1 = 2p 3 (2.15)

Exercise 2.1 Prove Eq.2.14.

Exercise 2.2 Show that {f, f } = 0 for any f.

Exercise 2.3 Assume that n is a positive integer. a) Evaluate {x 1 , pn 1 } b) Evaluate {xn 2 , p 2 }

Exercise 2.4 Verify Eq.2.5.

Exercise 2.5 Evaluate {3 + x 1 p^22 , p 1 x^22 p 3 }.

Exercise 2.6 Show that the Poisson bracket is not associative by giving a counter example.

So far, we defined the Poisson brackets of polynomials in the positions and momenta of one point mass only. Let us now consider the general case of a system of n point masses, m(r)^ with position vectors ~x(r)^ = (x( 1 r ), x( 2 r ), x( 3 r )) and momentum vectors ~p(r)^ =

(p( 1 r ), p( 2 r ), p( 3 r )), where r ∈ { 1 , 2 , ..., n}. How can we evaluate the Poisson brackets of expressions that involve all those positions and momentum variables? To this end, we need to define what the Poisson brackets in between positions and momenta of different particles should be. They are defined to be simply zero. Therefore, to summarize, we define the basic Poisson brackets of n masses as

{x( ir ), p( js )} = δi,j δr,s (2.16) {x( ir ), x( js )} = 0 (2.17) {p( ir ), p( js )} = 0 (2.18) (^4) If the product rule already reminds you of the product rule for derivatives (i.e., the Leibniz rule) this is not an accident. As we will see, the Poisson bracket can in fact be viewed as a sophisticated generalization of the notion of derivative.