Quantum Theory: Mathematics and Reality

Larry Landau

Mathematics Department, King's College London

Talk given at the King's College History of Mathematics Summer School, June 27-28 2001

1  Pure and Applied Mathematics

There are two aspects to mathematics: proof and truth, and we could (although many mathematicians might not share this viewpoint) make these two aspects correspond to two fundamental strands of mathematics: pure and applied. Pure mathematics concerns the intrinsic nature of mathematics itself, as an independent discipline. Applied mathematics concerns mathematics as a tool used in other disciplines, such as physics, chemistry, biology, engineering, finance, and so on. We could then take the view that pure mathematics deals with proof within a formal system with precise rules, and applied mathematics deals with truth in the real world. To the extent that we consider numbers to be real concrete things (or more precisely, a property of collections of real concrete things), we could think of number theory as mathematics applied to numbers, that is, as a branch of applied mathematics. To clarify this point, suppose we think of number as a property of bags of marbles, the number of marbles in the bag being a property of that collection of marbles. Then addition of two numbers yields the number of marbles obtained when the contents of two bags of marbles are placed in a third empty bag. We can then check how many marbles are in the bag and so we can check the truth of a statement such as 3+4 = 7. On the other hand, we could also prove that 3+4 = 7 by following calculational rules or using logical deduction from axioms about numbers.

To understand the nature of pure mathematics, we should consider what we do when we are doing mathematics. We are writing line after line of symbols on paper, the last line being the conclusion of the calculation, or theorem proved. If we take the view that this is the essence of mathematics (although some mathematicians may not take such a view), then we see that mathematics is a logical activity, based on axioms and rules of deduction. The meaning of the symbols is not so important, only how these symbols are used in the deduction. Thus mathematicians may interpret the symbols as meaning different things, yet they may agree that the mathematics itself is correctly carried out. According to this view, pure mathematics is a formal system of deduction, following precisely set out rules (which could, for example, be carried out by a computer) without an interpretation, and hence without a notion of truth. This is why in 1901 Bertrand Russell wrote

Mathematics may be defined as the subject in which we never know what we are talking about nor whether what we are saying is true.

In applied mathematics, on the other hand, the symbols which appear in the mathematics are interpreted according to the application. For example, in physics there are physical quantities such as position, velocity, acceleration, mass, energy, force and so on. Relations between these physical quantities and the values they take are expressed mathematically, and the symbols occuring in the mathematics are interpreted as these physical quantities.

2  Reality and Measurement

Newton's Absolute and Relative Time and Space

1. Absolute, true, and mathematical time, of itself and from its own nature, flows equably without relation to anything external; relative, apparent, and common time is some measure of duration by the means of motion, which is commonly used instead of true time, such as an hour, a day, a month, a year.
2. Absolute space, in its own nature, without relation to anything external, remains always similar and immovable. Relative space is some movable dimension which our senses determine by the position to bodies and which is commonly taken for immovable space.
3. Place is a part of space which a body takes up and is, according to the space, either absolute or relative.
4. Absolute motion is the translation of a body from one absolute place into another, and relative motion the translation from one relative place into another.

Einstein's Relativity of Space-Time

The Definition of Time

If at a point A of space there is a clock, an observer at A can determine the time values of events in the immediate proximity of A by finding the positions of the hands which are simultaneous with these events. If there is at the point B of space another clock in all respects resembling the one at A, it is possible for an observer at B to determine the time values of events in the immediate neighbourhood of B.

We cannot attach any absolute signification to the concept of simultaneity, but that two events which, viewed from a system of coordinates, are simultaneous, can no longer be looked upon as simultaneous events when envisaged from a system which is in motion relatively to that system.

In a similar way, space is defined by using a measuring rod, different observers using different measuring rods.

Here we see the principle clearly stated that our description of reality should be based not on absolute concepts, such as absolute time and absolute space, but on measurements by observers. Pictures of reality are replaced by observations of reality: the observer and the apparatus playing an important role in the description.

3  The Matrix Mechanics of Heisenberg

The present paper seeks to establish a basis for theoretical quantum mechanics founded exclusively upon relationships between quantities which in principle are observable.

So Heisenberg is developing a theory of the relationship between measurement results as opposed to developing a picture of reality.

The main observations of atoms at the time of the development of quantum theory concerned the spectral lines, light of definite frequencies given off by the atoms. Each type of atom had its own unique set of spectral lines. So Heisenberg tried to develop an atomic theory based on these spectral frequencies. A quantity x(t) in Heisenberg's quantum theory was represented by a collection of amplitudes and frequencies:

While walking back to Einstein's home after a seminar presented by Heisenberg in Berlin in the spring of 1926, Heisenberg told Einstein[3]:

We cannot observe electron orbits inside the atom, but the radiation which an atom emits during discharges enables us to deduce the frequencies and corresponding amplitudes of its electrons. Now, since a good theory must be based on directly observable magnitudes, I thought it more fitting to restrict myself to these, treating them, as it were, as representatives of the electron orbits.

Heisenberg was completely taken aback by Einstein's reply:

But you don't seriously believe that none but observable magnitudes must go into a physical theory?

Heisenberg answered:

Isn't that precisely what you have done with relativity? After all, you did stress the fact that it is impermissible to speak of absolute time, simply because absolute time cannot be observed; that only clock readings are relevant to the determination of time.

Einstein said:

You are moving on very thin ice. For you are suddenly speaking about what we know about nature and no longer about what nature really does. In science we ought to be concerned solely with what nature does. It might very well be that you and I know quite different things about nature. But who would be interested in that?

4  The Wave Mechanics of Schrödinger

de Broglie's Matter Waves

5  Wave-Particle Duality

6  EPR

1. a₁,a₂, or
2. a₁,b₂, or
3. b₁,a₂, or
4. b₁,b₂.

This experimental setup leads to some of the deepest questions in science, and has resulted in experimental results which seem to go against hundreds of years of science which had led to a deterministic ``clockwork'' picture of the universe. Perhaps in his thinking about the EPR experimental setup, here for once Einstein got it wrong. John S. Bell wrote a paper On the Einstein-Podolsky-Rosen paradox, Physics 1 (1964), which uncovered the mystery I'm going to tell you about. Using only basic ideas of probability, I'll show you the conventional proof that the universe is not local and deterministic. (There may be a way around this proof, perhaps involving a reinterpretation of the quantum formalism. But at present, no satisfactory way out has been found.) Here is an example of the reaction of some physicists to this argument:

Bell's argument deserves, and will receive, the closest scrutiny the human mind is capable of bringing to bear on it. If it survives that scrutiny, and if the experimental result is confirmed by others, then this will surely go down as one of the most incredible intellectual achievements in the history of science, and my own work will lie in ruins. I wish John von Neumann were here to see it.

E.T.Jaynes

7  Bell's Argument

For simplicity, we'll suppose that the measurement a₁ can only take two possible values, +1 and -1, and the same for the other measurements b₁,a₂,b₂. If I choose to measure a₁ in region R₁ and a₂ in region R₂, then I'll get a value for a₁ and a value for a₂ in each experimental trial. For example, I might get this experimental data:

That average, for a very large number of trials, is called the expectation value or mean value of the product a₁a₂ and is denoted áa₁a₂ñ. So if a very large number of trials yields the average 0.2, we would write áa₁a₂ñ = 0.2.

If, on the other hand, we choose to measure a₁ and b₂, we could work out the expectation value of the product: áa₁b₂ñ. Choosing to perform the other experiments, we could experimentally measure áb₁a₂ñ, and áb₁b₂ñ.

In summary, by doing four different experiments we can determine four expectation values áa₁a₂ñ, áa₁b₂ñ, áb₁a₂ñ, and áb₁b₂ñ. Now let's combine these four experimental results into one number, which we'll call the Bell number:

That's all we can say in general. But now let's see what effect locality and determinism have on the possible values of the Bell number.

1. The first hypothesis that we'll make is that in each experimental trial, the outcome of any experiment which I might choose to perform is predetermined. So, in a particular trial, if I decide to measure a₁, then the value which is measured was predetermined. Similarly, if I had decided to measure b₁, or a₂, or b₂, all the values are predetermined. Let's use + to denote the value +1 and - to denote the value -1. Then +-+ denotes the case where the predetermined value of a₁ (if that is what I choose to measure) is +1, the predetermined value of b₁ is -1, the predetermined value of a₂ is -1, and the predetermined value of b₂ is +1.
2. There are 16 cases for predetermined values: ++++, +++-, ++-+, ..., -- . We could label these cases with a parameter l = 1,2,¼,16. So for example, l = 3 might correspond to the case ++-+, and so on. So in each trial, the parameter l will take a particular value and this determines the value observed in a measurement of one of a₁,b₁,a₂ or b₂. Let's assign a probability to each possible value of l. So, for example, if the probability that l = 3 is 0.2, then in a long series of trials, a fraction 2/10 will have the value l = 3.
3. Now suppose in some trial I measure both a₁ and b₁. And suppose that in that trial l = 3, i.e. we have the case ++-+. Can we conclude that I will measure a₁ = +1 and b₁ = +1? NO. The reason is that the measurement of b₁ might cause a disturbance which would change the value of a₁ from what we would have observed if we had not measured b₁. The same argument applies had we measured both a₂ and b₂. There will still be a predetermined value for a₁ and b₁ in such a joint measurement (if the apparatus could be set up to do the joint measurement), but the predetermined values are not necessarily the same as for a single measurement.
4. But what happens if we measure a₁ and a₂ in a particular trial. Here we will use the locality postulate that no disturbance can travel faster than the speed of light (no action-at-a-distance). So if the space-time regions R₁ and R₂ are causally separated, the predetermined value for the outcome of a measurement of a₁ in region R₁ will be unaffected no matter what is done in region R₂. Consequently, we can say that if l = 3 and a joint measurement of a₁ and a₂ is made, then a₁ = +1 and a₂ = -1.
5. In summary, if the universe is deterministic and also local, then we can describe what happens in an EPR experiment like this. In each trial the parameter l takes a particular value, and this value determines the outcome of a measurement of a₁, of b₁, of a₂, and of b₂. So let's think of a₁ as a function of l: a₁(l), and similarly b₁(l), a₂(l), and b₂(l). Furthermore, because of locality, these outcomes are the same in a joint measurement of a₁ and a₂, or a₁ and b₂, or b₁ and a₂, or b₁ and b₂. So, for a given value of l, the product a₁a₂ takes the value a₁(l)a₂(l). In a long sequence of trials, the expectation value of the product a₁a₂ is then given by the very important formula

   áa1a2ñ = å l  a1(l)a2(l)P(l)

   where P(l) is the probability of the value l. Remember that the structure of this formula is a consequence of both determinism and locality.

   In the same way we have the formulas

   áa1b2ñ = å l  a1(l)b2(l)P(l) áb1a2ñ = å l  b1(l)a2(l)P(l) áb1b2ñ = å l  b1(l)b2(l)P(l)

6. So let's now put all the preceding formulas together to obtain a formula for the Bell number:

   bell = å l  a1(l)a2(l)P(l)+ å l  a1(l)b2(l)P(l)+ å l  b1(l)a2(l)P(l)- å l  b1(l)b2(l)P(l) = å l  [a1(l)a2(l)+a1(l)b2(l)+b1(l)a2(l)-b1(l)b2(l)]P(l) = å l  c(l)P(l) (3)

   where

   c(l) = a1(l)[a2(l)+b2(l)]+b1(l)[a2(l)-b2(l)]

   (4)

7. The two equations (3) and (4) are very important and will lead to a different conclusion from equation (2). From the formula (4), we conclude that c(l) takes only two possible values: ±2. To see this, suppose for example that a₁(l) = +1 and b₁(l) = -1. Then c(l) = [a₂(l)+b₂(l)]-[a₂(l)-b₂(l)] = 2b₂(l) = ±2. Suppose instead that a₁(l) = -1 and b₁(l) = -1. Then c(l) = -[a₂(l)+b₂(l)]-[a₂(l)-b₂(l)] = -2a₂(l) = ±2. The same conclusion holds for the other possible values to a₁(l) and b₁(l).
8. Since c(l) can only be +2 or -2, it follows that

   å l  c(l)P(l) £ å l  (+2)P(l) = 2 å l  P(l) = 2

   and

   å l  c(l)P(l) ³ å l  (-2)P(l) = -2 å l  P(l) = -2

   where we have used

   å l  P(l) = 1

   In summary, we have shown

   -2 £ å l  c(l)P(l) £ 2

   That is,

   -2 £ bell £ 2

   (5)

Equation (5) is the very important conclusion to our discussion based on local determinism. The general equation (2) has been replaced by the equation (5).

8  What Does Quantum Theory Say About the Bell Number?

How can we apply these ideas to the study of the Bell number? Since measurements in the space-time region R₁ do not disturb measurements in R₂, it follows that a₁ and a₂ are compatible, so [a₁,a₂] = 0. In the same way, we deduce all these relations:

However, measurements carried out in the region R₁ will in general disturb each other, so a₁ and b₁ will in general be complementary rather than compatible. Thus

We're now ready to apply this ``quantum algebra'' to the study of the Bell number. We can express the Bell number (1) in quantum theory in a way similar to what we did earlier:

But using equation (10) we can say more about how the prediction of local determinism is violated in quantum theory. Suppose we reconsider the preceding argument, but we simply interchange the roles of a₂ and b₂. Then instead of obtaining the quantity c (9), we would get the quantity d:

The inequality (15) was first found by B.Cirel'son [2], whose paper was smuggled out of the Soviet Union because at that time he was unable to leave. Today Cirel'son is in Tel Aviv, Israel.

9  Experimental Measurements of the Bell Number

References

[1]

A.Aspect, J.Dalibard, and G.Roger, Experimental test of Bell's inequalities using time-varying analyzers, Phys. Rev. Lett. 49 (1982) p.1804

[2]

B.S.Cirel'son, Lett. Math. Phys. 4 (1980) p.93

[3]

W.Heisenberg, Physics and Beyond (George Allen & Unwin Ltd, London) 1971, p.63

[4]

L.J.Landau, On the Violation of Bell's Inequality in Quantum Theory, Physics Letters A 120 (1987) pp.54-56

[5]

H.S.Thayer and J.H.Randall, Newton's Philosophy of Nature (Hafner Publishing Company, New York) 1953, p. 17

Footnotes:

¹ You could show (14) like this: