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1 The Transactional Interpretation of Quantum Mechanics John G. Cramer John G. Cramer Professor of Physics Department of Physics University of Washington Seattle, Washington, USA Presented at Georgetown University Washington, D.C. October 2, 2000
2 Recent Research at RHIC γ = 60 β = RHIC Au + Au collision at 130 Gev/nucleon measured with the STAR time projection Chamber on June 24, Colllisions may resemble the 1 st microsecond of the Big Bang. γ = 60 β =
3 Outline What is Quantum Mechanics? What is an Interpretation? Example: F = m a Listening to the formalism Lessons from E&M Maxwell s Wave Equation Wheeler-Feynman Electrodynamics & Advanced Waves The Transactional Interpretation of QM The Logic of the Transactional Interpretation The Quantum Transactional Model Paradoxes: 1. The Quantum Bubble 2. Schrödinger s Cat 3. Wheeler s Delayed Choice 4. The Einstein-Podolsky-Rosen Paradox Application of TI to Quantum Experiments Conclusion
4 Theories and Interpretations
5 What is Quantum Mechanics? Quantum mechanics is a theory that is our current standard model for describing the behavior of matter and energy at the smallest scales (photons, atoms, nuclei, quarks, gluons, leptons, ). Like all theories, it consists of a mathematical formalism and an interpretation of that formalism. However, while the formalism has been accepted and used for 75 years, its interpretation remains a matter of controversy and debate, and there are several rival interpretations on the market.
6 Example of an Interpretation: Formalism: F = m a Newton s 2 nd Law
7 Example of an Interpretation: Formalism: F = m a Newton s 2 nd Law Interpretation: The vector force on a body is proportional to the product of its scalar mass, which is positive, and the 2 nd time derivative of its vector position.
8 Example of an Interpretation: Formalism: F = m a Newton s 2 nd Law Interpretation: The vector force on a body is proportional to the product of its scalar mass, which is positive, and the 2 nd time derivative of its vector position. What this Interpretation does: It relates the formalism to physical observables It avoids paradoxes that arise when m<0. It insures that F a.
9 What is an Interpretation? The interpretation of a formalism should: Provide links between the mathematical symbols of the formalism and elements of the physical world;
10 What is an Interpretation? The interpretation of a formalism should: Provide links between the mathematical symbols of the formalism and elements of the physical world; Neutralize the paradoxes; all of them;
11 What is an Interpretation? The interpretation of a formalism should: Provide links between the mathematical symbols of the formalism and elements of the physical world; Neutralize the paradoxes; all of them; Provide tools for visualization or for speculation and extension.
12 What is an Interpretation? The interpretation of a formalism should: Provide links between the mathematical symbols of the formalism and elements of the physical world; Neutralize the paradoxes; all of them; Provide tools for visualization or for speculation and extension. It should not make its own testable predictions! It should not have its own sub-formalism!
13 Listening to the Formalism of Quantum Mechanics Consider a quantum matrix element: <S> = v ψ S ψ dr 3 = <f S i> a ψ* - ψ sandwich. What does this suggest?
14 Listening to the Formalism of Quantum Mechanics Consider a quantum matrix element: <S> = v ψ S ψ dr 3 = <f S i> a ψ* - ψ sandwich. What does this suggest? Hint: The complex conjugation in ψ is the Wigner operator for time reversal.
15 Listening to the Formalism of Quantum Mechanics Consider a quantum matrix element: <S> = v ψ S ψ dr 3 = <f S i> a ψ* - ψ sandwich. What does this suggest? Hint: The complex conjugation in ψ is the Wigner operator for time reversal. If ψ is a retarded wave, then ψ is an advanced wave.
16 Listening to the Formalism of Quantum Mechanics Consider a quantum matrix element: <S> = v ψ S ψ dr 3 = <f S i> a ψ* - ψ sandwich. What does this suggest? Hint: The complex conjugation in ψ is the Wigner operator for time reversal. If ψ is a retarded wave, then ψ is an advanced wave. If ψ = Α e i(kr-ω t) then ψ = Α e i(-kr+ω t) (retarded) (advanced)
17 Lessons from Classical E&M
18 Maxwell s Electromagnetic Wave Equation 2 F i = 1/c 2 2 F i / t 2 This is a 2 nd order differential equation, which has two time solutions, retarded and advanced.
19 Maxwell s Electromagnetic Wave Equation 2 F i = 1/c 2 2 F i / t 2 This is a 2 nd order differential equation, which has two time solutions, retarded and advanced. Conventional Approach: Choose only the retarded solution (a causality boundary condition).
20 Maxwell s Electromagnetic Wave Equation 2 F i = 1/c 2 2 F i / t 2 This is a 2 nd order differential equation, which has two time solutions, retarded and advanced. Conventional Approach: Choose only the retarded solution (a causality boundary condition). Wheeler-Feynman Approach: Use ½ retarded and ½ advanced (time symmetry).
21 Lessons from Wheeler-Feynman Absorber Theory
22 A Classical Wheeler-Feynman Electromagnetic Transaction The emitter sends retarded and advanced waves. It offers to transfer energy.
23 A Classical Wheeler-Feynman Electromagnetic Transaction The emitter sends retarded and advanced waves. It offers to transfer energy. The absorber responds with an advanced wave that confirms the transaction.
24 A Classical Wheeler-Feynman Electromagnetic Transaction The emitter sends retarded and advanced waves. It offers to transfer energy. The absorber responds with an advanced wave that confirms the transaction. The loose ends cancel and disappear, and energy is transferred.
25 The Transactional Interpretation of Quantum Mechanics
26 The Logic of the Transactional Interpretation 1. Interpret Maxwell s wave equation as a relativistic quantum wave equation (for m rest = 0).
27 The Logic of the Transactional Interpretation 1. Interpret Maxwell s wave equation as a relativistic quantum wave equation (for m rest = 0). 2. Interpret the relativistic Klein-Gordon and Dirac equations (for m rest > 0)
28 The Logic of the Transactional Interpretation 1. Interpret Maxwell s wave equation as a relativistic quantum wave equation (for m rest = 0). 2. Interpret the relativistic Klein-Gordon and Dirac equations (for m rest > 0) 3. Interpret the Schrödinger equation as a nonrelativistic reduction of the K-G and Dirac equations (for m rest > 0).
29 Step 1: The emitter sends out an offer wave Ψ. The Quantum Transactional Model
30 Step 1: The emitter sends out an offer wave Ψ. The Quantum Transactional Model Step 2: The absorber responds with a confirmation wave Ψ*.
31 Step 1: The emitter sends out an offer wave Ψ. The Quantum Transactional Model Step 2: The absorber responds with a confirmation wave Ψ*. Step 3: The process repeats until energy and momentum is transferred and the transaction is completed (wave function collapse).
32 The Transactional Interpretation and Wave-Particle Duality The completed transaction projects out only that part of the offer wave that had been reinforced by the confirmation wave. Therefore, the transaction is, in effect, a projection operator. This explains wave-particle duality.
33 The Transactional Interpretation and the Born Probability Law Starting from E&M and the Wheeler- Feynman approach, the E-field echo that the emitter receives from the absorber is the product of the retarded-wave E-field at the absorber and the advancedwave E-field at the emitter.
34 The Transactional Interpretation and the Born Probability Law Starting from E&M and the Wheeler- Feynman approach, the E-field echo that the emitter receives from the absorber is the product of the retarded-wave E-field at the absorber and the advancedwave E-field at the emitter. Translating this to quantum ψψ* mechanical terms, the echo that the emitter receives from each potential absorber is ψψ*, leading to the Born Probability Law. ψ
35 The Role of the Observer in the Transactional Interpretation In the Copenhagen interpretation, observers have a special role as the collapsers of wave functions. This leads to problems, e.g., in quantum cosmology where no observers are present.
36 The Role of the Observer in the Transactional Interpretation In the Copenhagen interpretation, observers have a special role as the collapsers of wave functions. This leads to problems, e.g., in quantum cosmology where no observers are present. In the transactional interpretation, transactions involving an observer are the same as any other transactions.
37 The Role of the Observer in the Transactional Interpretation In the Copenhagen interpretation, observers have a special role as the collapsers of wave functions. This leads to problems, e.g., in quantum cosmology where no observers are present. In the transactional interpretation, transactions involving an observer are the same as any other transactions. Thus, the observer-centric aspects of the Copenhagen interpretation are avoided.
38 Quantum Paradoxes
39 Paradox 1: The Quantum Bubble Situation: A photon is emitted from an isotropic source.
40 Paradox 1: The Quantum Bubble Situation: A photon is emitted from an isotropic source. Question (Albert Einstein): If a photon is detected at Detector A, how does the photon s wave function at the location of Detectors B & C know that it should vanish?
41 Paradox 1: The Quantum Bubble Situation: A photon is emitted from an isotropic source. Question (Albert Einstein): If a photon is detected at Detector A, how does the photon s wave function at the location of Detectors B & C know that it should vanish?
42 Paradox 1: Application of the Transactional Interpretation to the Quantum Bubble A transaction develops between the source and detector A, transferring the energy there and blocking any similar transfer to the other potential detectors, due to the 1-photon boundary condition. The transactional handshakes acts nonlocally to answer Einstein s question. This is an extension of Pilot-Wave idea of debroglie.
43 Experiment: A cat is placed in a sealed box containing a device that has a 50% probability of killing the cat. Paradox 2: Schrödinger s Cat
44 Paradox 2: Schrödinger s Cat Experiment: A cat is placed in a sealed box containing a device that has a 50% probability of killing the cat. Question 1: When does the wave function collapse? What is the wave function of the cat just before the box is opened? (Ψ = ½ dead + ½ alive?)
45 Paradox 2: Schrödinger s Cat Experiment: A cat is placed in a sealed box containing a device that has a 50% probability of killing the cat. Question 1: When does the wave function collapse? What is the wave function of the cat just before the box is opened? (Ψ = ½ dead + ½ alive?) Question 2: If we observe Schrödinger, what is his wave function during the experiment? When does it collapse?
46 Paradox 2: Application of the Transactional Interpretation to Schrödinger s Cat A transaction either develops between the source and the detector, or else it does not. If it does, the transaction forms nonlocally, not at some particular time. Therefore, asking when the wave function collapsed was asking the wrong question.
47 Paradox 3: Wheeler s Delayed Choice A source emits one photon. Its wave function passes through two slits, producing interference.
48 Paradox 3: Wheeler s Delayed Choice A source emits one photon. Its wave function passes through two slits, producing interference. The observer can choose to either: (a) measure the interference pattern (wavelength) at E
49 Paradox 3: Wheeler s Delayed Choice A source emits one photon. Its wave function passes through two slits, producing interference. The observer can choose to either: (a) measure the interference pattern (wavelength) at E or (b) measure the slit position with telescopes T 1 and T 2.
50 Paradox 3: Wheeler s Delayed Choice A source emits one photon. Its wave function passes through two slits, producing interference. The observer can choose to either: (a) measure the interference pattern (wavelength) at E or (b) measure the slit position with telescopes T 1 and T 2. He decides which to do after the photon has passed the slits.
51 Paradox 3: Application of the Transactional Interpretation If plate E is up, a transaction forms between E and the source S and involves waves passing through both slits.
52 Paradox 3: Application of the Transactional Interpretation If plate E is up, a transaction forms between E and the source S and involves waves passing through both slits. If the plate E is down, a transaction forms between telescope T 1 or T 2 and the source S, and involves waves passing through only one slit.
53 Paradox 3: Application of the Transactional Interpretation If plate E is up, a transaction forms between E and the source S. If the plate E is down, a transaction forms between one of the telescopes (T 1, T 2 ) and the source S. In either case, when the decision was made is irrelevant.
54 Paradox 4: EPR Experiments Malus and Furry An EPR Experiment measures the correlated polarizations of a pair of entangled photons, obeying Malus Law [P(θ rel ) = Cos 2 θ rel ]
55 Paradox 4: EPR Experiments Malus and Furry An EPR Experiment measures the correlated polarizations of a pair of entangled photons, obeying Malus Law [P(θ rel ) = Cos 2 θ rel ] The measurement gives the same result as if both filters were in the same arm.
56 Paradox 4: EPR Experiments Malus and Furry An EPR Experiment measures the correlated polarizations of a pair of entangled photons, obeying Malus Law [P(θ rel ) = Cos 2 θ rel ] The measurement gives the same result as if both filters were in the same arm. Furry proposed to place both photons in the same random polarization state. This gives a different and weaker correlation.
57 Paradox 4: Application of the Transactional Interpretation to EPR An EPR experiment requires a consistent double advancedretarded handshake between the emitter and the two detectors.
58 Paradox 4: Application of the Transactional Interpretation to EPR An EPR experiment requires a consistent double advancedretarded handshake between the emitter and the two detectors. The lines of communication are not spacelike but negative and positive timelike. While spacelike communication has relativity problems, timelike communication does not.
59 Faster Than Light?
60 Is FTL Communication Possible with EPR Nonlocality? Question: Can the choice of measurements at D 1 telegraph information as the measurement outcome at D 2?
61 Is FTL Communication Possible with EPR Nonlocality? Question: Can the choice of measurements at D 1 telegraph information as the measurement outcome at D 2? Answer: No! Operators for measurements D 1 and D 2 commute. [D 1, D 2 ]=0. Choice of measurements at D 1 has no observable consequences at D 2. (Eberhard s Theorem)
62 Is FTL Communication Possible with EPR Nonlocality? Question: Can the choice of measurements at D 1 telegraph information as the measurement outcome at D 2? Answer: No! Operators for measurements D 1 and D 2 commute. [D 1, D 2 ]=0. Choice of measurements at D 1 has no observable consequences at D 2. (Eberhard s Theorem) Levels of EPR Communication: 1. Enforce conservation laws (Yes)
63 Is FTL Communication Possible with EPR Nonlocality? Question: Can the choice of measurements at D 1 telegraph information as the measurement outcome at D 2? Answer: No! Operators for measurements D 1 and D 2 commute. [D 1, D 2 ]=0. Choice of measurements at D 1 has no observable consequences at D 2. (Eberhard s Theorem) Levels of EPR Communication: 1. Enforce conservation laws (Yes) 2. Talk observer-to-observer (No!) [Unless nonlinear QM?!)
64 Conclusions (Part 1) The Transactional Interpretation is visible in the quantum formalism It involves fewer independent assumptions than its alternatives. It solves the quantum paradoxes; all of them. It explains wave-function collapse, waveparticle duality, and nonlocality. ERP communication FTL is not possible!
65 Application: An Interaction-Free Measurement
66 Elitzur-Vaidmann Interaction-Free Measurements Suppose you are given a set of photon-activated bombs, which will explode when a single photon touches their optically sensitive triggers.
67 Elitzur-Vaidmann Interaction-Free Measurements Suppose you are given a set of photon-activated bombs, which will explode when a single photon touches their optically sensitive trigger. However, some fraction of the bombs are duds which will freely pass an incident photon without exploding.
68 Elitzur-Vaidmann Interaction-Free Measurements Suppose you are given a set of photon-activated bombs, which will explode when a single photon touches their optically sensitive triggers. However, some fraction of the bombs are duds which will freely pass an incident photon without exploding. Your assignment is to sort the bombs into live and dud categories. How can you do this without exploding all the live bombs?
69 Elitzur-Vaidmann Interaction-Free Measurements Suppose you are given a set of photon-activated bombs, which will explode when a single photon touches their optically sensitive trigger. However, some fraction of the bombs are duds which will freely pass an incident photon without exploding. Your assignment is to sort the bombs into live and dud categories. How can you do this without exploding all the live bombs? Classically, the task is impossible. All live bombs explode!
70 Elitzur-Vaidmann Interaction-Free Measurements Suppose you are given a set of photon-activated bombs, which will explode when a single photon touches their optically sensitive triggers. However, some fraction of the bombs are duds which will freely pass an incident photon without exploding. Your assignment is to sort the bombs into live and dud categories. How can you do this without exploding all the live bombs? Classically, the task is impossible. All live bombs explode! However, using quantum mechanics, you can do it!
71 The Mach-Zender Interferometer A Mach-Zender intereferometer splits a light beam at S 1 into two paths, A and B, having equal lengths, and recombines the beams at S 2. All the light goes to detector D 1 because the beams interfere destructively at detector D 2.
72 The Mach-Zender Interferometer A Mach-Zender intereferometer splits a light beam at S 1 into two paths, A and B, having equal lengths, and recombines the beams at S 2. All the light goes to detector D 1 because the beams interfere destructively at detector D 2. D 1 : L S 1 r Ar S 2 t D 1 and L S 1 t Br S 2 r D 1 => in phase
73 The Mach-Zender Interferometer A Mach-Zender intereferometer splits a light beam at S 1 into two paths, A and B, having equal lengths, and recombines the beams at S 2. All the light goes to detector D 1 because the beams interfere destructively at detector D 2. D 1 : L S 1 r Ar S 2 t D 1 and L S 1 t Br S 2 r D 1 => in phase D 2 : L S 1 r Ar S 2 r D 2 and L S 1 t Br S 2 t D 2 => out of phase
74 A M-Z Inteferometer with an Opaque Object in Beam A If an opaque object is placed in beam A, the light on path B goes equally to detectors D 1 and D 2.
75 A M-Z Inteferometer with an Opaque Object in Beam A If an opaque object is placed in beam A, the light on path B goes equally to detectors D 1 and D 2. This is because there is now no interference, and splitter S 2 divides the incident light equally between the two detector paths.
76 A M-Z Inteferometer with an Opaque Object in Beam A If an opaque object is placed in beam A, the light on path B goes equally to detectors D 1 and D 2. This is because there is now no interference, and splitter S 2 divides the incident light equally between the two detector paths. Therefore, detection of a photon at D 2 (or an explosion) signals that a bomb has been placed in path A.
77 How to Sort the Bombs Send in a photon with the bomb in A. If it is a dud, the photon will always go to D 1. If it is a live bomb, ½ of the time the bomb will explode, ¼ of the time it will go to D 1 and ¼ of the time to D 2.
78 How to Sort the Bombs Send in a photon with the bomb in A. If it is a dud, the photon will always go to D 1. If it is a live bomb, ½ of the time the bomb will explode, ¼ of the time it will go to D 1 and ¼ of the time to D 2. Therefore, on each D 1 signal, send in another photon. On a D 2 signal, stop, you have a live bomb! After 10 or so D 1 signals, stop, you have a dud bomb! By this process, you will find unexploded 1/3 of the live bombs and will explode 2/3 of the live bombs.
79 Quantum Knowledge Thus, we have used quantum mechanics to gain a kind of knowledge (i.e., which unexploded bombs are live) that is not accessible to us classically. or
80 Quantum Knowledge Thus, we have used quantum mechanics to gain a kind of knowledge (i.e., which unexploded bombs are live) that is not accessible to us classically. Further, we have detected the presence of an object (the live bomb), without a single photon having interacted with that object. Only the possibility of an interaction was required for the measurement. or
81 Quantum Knowledge Thus, we have used quantum mechanics to gain a kind of knowledge (i.e., which unexploded bombs are live) that is not accessible to us classically. Further, we have detected the presence of an object (the live bomb), without a single photon having interacted with that object. Only the possibility of an interaction was required for the measurement. Q: How can we understand this curious quantum behavior? or
82 Quantum Knowledge Thus, we have used quantum mechanics to gain a kind of knowledge (i.e., which unexploded bombs are live) that is not accessible to us classically. Further, we have detected the presence of an object (the live bomb), without a single photon having interacted with that object. Only the possibility of an interaction was required for the measurement. Q: How can we understand this curious quantum behavior? A: Apply the transactional interpretation. or
83 Transactions for No Object There are two allowed paths between the light source L and the detector D 1.
84 Transactions for No Object There are two allowed paths between the light source L and the detector D 1. If the paths have equal lengths, the offer waves ψ to D 1 will interfere constructively, while the offer ψ waves to D 2 interfere destructively and cancel.
85 Transactions for No Object There are two allowed paths between the light source L and the detector D 1. If the paths have equal lengths, the offer waves ψ to D 1 will interfere constructively, while the offer ψ waves to D 2 interfere destructively and cancel. The confirmation waves ψ* traveling back to L along both paths back to L will confirm the transaction.
86 Transactions with Bomb Present (1) An offer wave from L on path A will reach the bomb. An offer wave on path B reaching S 2 will split equally, reaching each detector with ½ amplitude.
87 Transactions with Bomb Present (2) The bomb will return a confirmation wave on path A. Detectors D 1 and D 2 will each return confirmation waves, both to L and to the back side of the bomb. The amplitudes of the confirmation waves at L will be ½ from the bomb and ¼ from each of the detectors, and a transaction will form based on those amplitudes.
88 Transactions with Bomb Present (3) Therefore, when the bomb does not explode, it is nevertheless probed by virtual offer and confirmation waves from both sides. The bomb must be capable of interaction with these waves, even though no interaction takes place (because no transaction forms).
89 Application: The Quantum Xeno Effect
90 Quantum Xeno Effect Improvement of Interaction-Free Measurements Kwait, et al, have devised an improved scheme for interaction-free measurements that can have efficiencies approaching 100%. Their trick is to use the quantum Xeno effect to probe the bomb with weak waves many times. The incident photon runs around an optical racetrack N times, until it is deflected out.
91 Efficiency of the Xeno Interaction-Free Measurements If the object is present, the emerging photon at D H will be detected with a probability P D = Cos 2N (π/2n). The photon will interact with the object with a probability P R = 1 - P D = 1 - Cos 2N (π/2n). When N is large, P D 1 (π/2) 2 /N and P R (π/2) 2 /N. Therefore, the interaction probability decreases as 1/N.
92 Offer Waves with No Object in the V Beam This shows an unfolding of the Xeno apparatus when no object is present in the V beam. In this case the photon wave is split into horizontal (H) and vertical (V) components, and then recombined. The successive R filters each rotate the plane of polarization by π/2n. The photon emerges with V polarization.
93 Offer Waves with an Object in the V Beam This shows an unfolding of the Xeno apparatus when an object is present in the V beam. In this case the photon wave is repratedly reset to horizontal (H) polarization. The photon emerges with H polarization.
94 Confirmation Waves with an Object in the V Beam This shows the confirmation waves for an unfolding of the Xeno apparatus when an object is present in the V beam. In this case the photon wave is reset to horizontal (H) polarization. The wave returns to the source L with the H polarization of the initial offer wave.
95 Conclusions (Part 2) The Transactional Interpretation can account for the non-classical information provided by interaction-freemeasurements. The roles of the virtual offer and confirmation waves in probing the object being measured lends support to the transactional view of the process. The examples shows the power of the interpretation in dealing with counterintuitive quantum optics results.
96 Applications of the Transactional Interpretation of Quantum Mechanics John G. Cramer John G. Cramer Professor of Physics Department of Physics University of Washington Seattle, Washington, USA Presented at the Breakthrough Physics Lecture Series Marshall Space Flight Center Marshall, Alabama August 17, 2000
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