1. Lecture from Quantum Mechanics

3. Marek Zrałek Field Theory and Particle Physics Department. Silesian University Lecture 9 “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.” ― Albert Einstein

6. Hidden variables Quantum mechanics does not give a complete description of reality, microscopic physical systems are described by additional parameters: When in addition to all other parameters determining the quantum state in addition we know λ, we will obtain full information about our system, like in classical physics. Measurements of any observables in such a state will give a determined result. Objective reality When we measure observables A and B in distant points in space then: Local determinism, there is no transfer of interaction at a distance

7.  J. von Neumann (1932) - presented evidence that QM does not allow the existence of hidden variables.  D.Bohm (1952) - introduced a non-local hidden variables.  J.S.Bell (1964) Twierdzenie Bella – in some experiments, all local realistic theories of hidden variables are incompatible with QM  J.S.Bell (1966) - he showed that the assumptions of von Neumann were too strong, you can introduce non-local hidden variables, he thought, too, that may exist local hidden variables. On the Problem of Hidden Variables in Quantum Mechanics. John S. Bell, Rev.Mod. Phys.38 (1966)447..On the Einstein-Podolsky-Rosen paradox J.S.Bell,Physics,1 (1964)195;

8. Bell, J.S. [1964], “On the Einstein-Podolsky-Rosen paradox,” Physics, 1: 195–200.

9. There are a number of Bell's inequality, we will show the first one originally introduced by Bell in 1964. But first we give example of the theory of hidden variables (G. Greenstein and A. Zajonc; "The Quantum Challenge, Modern Research on the Foundations of Quantum Mechanics," Jones and Bartlett Publishers, 1997) We assume that the hidden variables imitate the classic notion of spin. Assume that the Stern – Gerlach device indicates a A = +1 or -1, which depends on the sign of cos [θ], namely A = sign[cos(θ)] And similarly B(θ b =180 – (θ-φ)) B = sign[cos(180 – (θ - φ))] = sign[ - cos( θ - φ)]. Then we obtain AB = sign[ - cos(θ) cos(θ - φ)] Hidden direction of spin "seen" by Alice θ The direction of spin "seen" by Bob

10. x -x θ A(θ) = +1 A(θ) = -1

11. is calculated in the way: The AB values for various range of angles We obtain: - You can see that there are a set of equipment for which MK does not agree with our adopted theory of hidden variables. But maybe we chose badly hidden parameters. But may be, it is possible to select hidden variables in such a way, that the result of QM will be satisfied. Bell showed that any choice will not give good results. cos( )

12. A(θ) = +1 A(θ) = -1 φ

13. Original Bell paper B (J.Bell, "On the Einstein-Podolsky-Rosen paradox", Physics, 1 (1964) 195 ) concerned the average correlation value E (a, b),... for the three directions a, b, c. Later Bell work ("On the problem of a hidden variables in quantum mechanics," Rev. Mod. Phys., 38 (1966) 447) as well as others authors (S.Kochen, E.P.Specker, "The issue on hidden variables in quantum mechanics", J. Math. Phys., 17 (1967) J.F.Clauser, A.Shimony, Bell's theorem: experimental and test applications '' Rep. Threshold. Phys., 41 (1978) 1881, A.Shimony, A.Zeilinger, "Bell's theorem without Inequalities", 58 (1990) 1131 and many others), they gave similar reports suitable for direct experimental testing. Currently, there are a number of such relations generally called Bell inequalities. We give Bell arguments from the first paper: there are three directions described vectors, Bell showed that the assumptions of a local determinism and objective reality leads to inequalities: which for certain settings of directions a, b, c is satisfied, for example. a = b = c: and for others it does not: 60 0

15. Bell’s argumentation: The results of measurements performed by Alice and Bob only depend on the apparatus settings and unspecified local properties of the particles So, we have and not Expectation value of the product AB is equal: We normalize a hidden variables in such a way that: We would like to have : Due to the fact that the results of Alice and Bob are opposite, in the hidden variables language it mean that: This condition is necessary for: And we have:

16. Using the condition: We have the final result: In a similar way all other inequalities in theories with hidden variables are proved. So we have :

17. Until now, to show the impossibility of constructing a theory with hidden variables, we used observables which do not commute and assigning them potentially measured values, regardless of whether we make any measurement or not (eg. A spin ½ on-axis "x" axis and the "z"). Non-existence of hidden variables can be shown also for observables which are compatible. In the "classical" world we can consider for example a hair color (A), growth (B) and weight (C), or hair color (A), shoe size (L) and sex (M). The result of "measurement" od A does not depend on whether, in addition we decide to measure a growth and weight or size of shoes and gender. In Quantum Mechanics it is no longer so. A measurement may depend on whether we additionally decide to measure B and C or L and M. At first Bell showed that (1966) and later Kochen and Specker (BKS theorem about contextuality). The prove can be find in the article of D. Mermin, Rev. of Mod. Phys. 65 (1993) 803 We show the contextuality on the example of a particle with spin 1. Three spin operators do non commute, but In addition for spin s =1 there is : Show that for any spin that squares of operators commute.

18. z x y x’x’ y’y’ x y z z x’ y’ 1 1 0 0 1 1 1 0 1 1 0 1 0 1 1 1 1 0 In a theory with hidden variables must exist unequivocal way to assign a value of 1 or 0 for each direction of the angular momentum, which is independent on the choice of coordinate axes. In the QM we can not select a squares of angular momentum in a unequivocal way. The value of depends in which direction we measure and. It's like hair colour depend on whether we have decided to observe a shoe size and sex or weight and height. We can measure the squares of angular momentum along any of the three axes:

19. D. Mermina, Rev. of Mod.Phys. 65 (1993) 803 J. Math. Phys., 17, (1968), 59

20. Number of citation of Bell's papers (from L.E. Ballentine, „Foudation of quantum mechanics since the Bell inequalities”, Am. J. Phys., 55(1987)785). The Aspect papers allowing to verify the Bell inequality „The most profound and shocking scientific discovery " „Bell's theorem is considered by many people as one of the deepest and most surprising discoveries of modern physics, concerning not only the interpretation of Quantum Mechanics, but also the perception of the whole of our world "

21. John Steward Bell (1928 – 1990)

22. Classical equivalent of the Bell inequalities: N.D.Mermin, "Is the moon there when nobody looks?", Physics Today, 38 (1985) 38. Three numbers: 1,2,3 Two colours Sends signals eg. balls And unknown codes in each balls Message sent to one device can not cause reactions which dependent on the position of the switch in the other device. Transferring ball does not know the status of the device. Sent balls have codes, informing which lamp should light up, depending on the setting numbers eg. (RGR). There are 8 possible codes {RRG,RGR,GRR,GRG,GGR,RGG, GGG,RRR}. We have 8x8 = 64 pairs of code carried by the balls. Codes carried by balls are not know, we observe only lighting lamps depending on the set of indicators No connection between equipments recording the codes

23. The course of events: The transmitter sends two balls with codes for the two receivers. Before the balls reach the device, the switches are set randomly in the position 1,2 or 3. Lamps light up only when the signal (ball) reaches them. The string is repeated many times, and the results can be e.g. such: Let us assume that the result of observation is the next:  When the switches are in the same position (11, 22, 33) always illuminates the same colour (half - red, half - green)  When you do not look at the switches setting (11,22,33,12,13,21,23,31,32), then in an equal number of lights come on green and red.  Let us try to find some codes on balls which give such results ???  We will show later that this may be in the real quantum experience

24. We ask, what code (one of 64) have the balls? Since for the same numbers, balls always appear in the same colour, this means that the codes on the balls must be the same. If it was not so, in some cases they lamps ignite in different colours of light. So we have 8 possibilities, let's examine them 11(zz), 12(zz), 13(zz), 21(zz), 22(zz), 23(zz), 31(zz), 32(zz), 33(zz) not possible 11(cc), 12(cc), 13(cc), 21(cc), 22(cc), 23(cc), 31(cc), 32(cc), 33(cc) not possible 11(zz), 12(zz), 13(zc), 21(zz), 22(zz), 23(zc), 31(cz), 32(cz), 33(cc) 11(zz), 12(zc), 13(zz), 21(cz), 22(cc), 23(cz), 31(zz), 32(zc), 33(zz) 11(cc), 12(cz), 13(zc), 21(zc), 22(zz), 23(zz), 31(zc), 32(zz), 33(zz) 11(zz), 12(zc), 13(zc), 21(cz), 22(cc), 23(cc), 31(zc), 32(cc), 33(cc) 11(cc), 12(cz), 13(cc), 21(zc), 22(zz), 23(cz), 31(cc), 32(cz), 33(cc) 11(cc), 12(cc), 13(cz), 21(cc), 22(cc), 23(cz), 31(zc), 32(zc), 33(zz)

25.  We see at first, that the two codes (ZZZ) (CCC) are not possible  In any other case, 5 times the colours of light are the same and in 4 situations the colour are different. The probability of ignition of lamps of the same colour = 5/9 But MK said, that the probability = 1/2 Let's see now, that the described situation at the micro - world level is a natural Objective reality which can explaining our observations does not exist, unless the device "contact" with each other (but it would be "Spooky interaction at a distance") Only one possibility is real, or (1) there is interaction at a distance, or (2) the reality that was awaited by Einstein, at the level of the micro-world does not exist

26. Alice Bob Spin 1/2 position 1 position 2 position 3 For ALICE In every direction - observation of the spin projection +1/2 corresponds to the green light while for spin -1/2 -- light is red Let us now consider various situations Alice observed in a 1 +1/2, the electron flying to Bob has -1/2 on a1, and this means: Bob observations : P(a 2 ) = (cos(60/2)) 2 = 3/4 P(-a 2 ) = (cos(120/2)) 2 =1/4 and the same for a 3. For one on the four cases the lamps have the same colour. Three times different colour of lights light up For BOB, vice versa; s = -1/2 s = +1/2

27. x y z u θ φ

28. Alice observed in a 1 -1/2, the electron flying to Bob has +1/2 on a 1, and this means: Bob observations : P(a 2 ) = (cos(120/2)) 2 =1/4 P(-a 2 ) = (cos(60/2)) 2 =3/4 and the same for a 3. So in the same way as before, one of the four cases the same colour of light is turned. Three times different colors of light appear up Let's collect now all possibilities Numbers of switches Colour of light Due to correlations spins are always opposite, but the colours are the same In each of these situations in three- quarters of cases, the lamps have different colors, and only in a one quarter of cases the same colors appear  Whenever Alice and Bob have the same numbers of switches, the lamps light up with the same colour  Probability that the lights have the same colour:  Probability that lamps have different colours:

29. Everything can be proved in the other way : because We see that there is a situation in the micro-world, which exactly agrees with our imaginary classic example. So, in another way you can see, that without action at a distance, we are not able (using the code carried by the balls = hidden variables associated with particles) to explain the described scenario. We have to accept that in the micro-world there is not: Local determinism and Objective reality. We will show that this scenario has been confirmed experimentally. Prove on seminar

30. Answer of experiments All experiments which check Bell inequalities were made for entangled spin states as proposed by D. Bohm (1951)  Experyments with protons: M.Lemehi-Rachti W. Mittig, „Quantum Mechanics and hidden variables”, Phys. Rev.14(1976)2543.  Experiments witt entangled fotons: L.R. Kasday,J.D. Ulman,C.S. Wu, „Angular correlation of Compton scattered annihilation photon and hidden variables”, Nuovo Cimento, 25B(1975)633. e + e - -> pozytronium -> 2 photons Too high photon energy, problems with their polarity  Later a good source of polarized photons was found: Calcium atoms have two electrons in the s state, out of the enclosed shell. Ground state J = 0: Not enough precision

31. Since 1982, we know that in the micro-world: there is no local determinism lack of objective reality  Now new experiments confirm the QM with accuracy 100σ Quantum non-locality There is no immediate transfer of interactions and so the STR it is not violated but it is The correlation at the distance = "Passion at a distance" Alain Aspect i John Bell (1985)