1. The Transactional Interpretation of Quantum Mechanics www
The Transactional Interpretation of Quantum Mechanics

2. Recent Research at RHIC
g = 60
b =
g = 60
b =
RHIC Au + Au
collision at
130 Gev/nucleon
measured with
the STAR
time projection
Chamber on
June 24, 2000.
Colllisions may
resemble the 1st microsecond of the Big Bang.

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:
The Quantum Bubble
Schrödinger’s Cat
Wheeler’s Delayed Choice
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: Newton’s 2nd Law
Formalism: F = m a

7. Example of an Interpretation: Newton’s 2nd Law
Formalism: F = m a
Interpretation: “The vector force on a body is proportional to the product of its scalar mass, which is positive, and the 2nd time derivative of its vector position.”

8. Example of an Interpretation: Newton’s 2nd Law
Formalism: F = m a
Interpretation: “The vector force on a body is proportional to the product of its scalar mass, which is positive, and the 2nd 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> = òv y* S y dr3 = <f | S | i>
… a y* - y “sandwich”. What does this suggest?

14. “Listening” to the Formalism of Quantum Mechanics
Consider a quantum matrix element:
<S> = òv y* S y dr3 = <f | S | i>
… a y* - y “sandwich”. What does this suggest?
Hint: The complex conjugation in y* is the Wigner operator for time reversal.

15. “Listening” to the Formalism of Quantum Mechanics
Consider a quantum matrix element:
<S> = òv y* S y dr3 = <f | S | i>
… a y* - y “sandwich”. What does this suggest?
Hint: The complex conjugation in y* is the Wigner operator for time reversal. If y is a retarded wave, then y* is an advanced wave.

16. “Listening” to the Formalism of Quantum Mechanics
Consider a quantum matrix element:
<S> = òv y* S y dr3 = <f | S | i>
… a y* - y “sandwich”. What does this suggest?
Hint: The complex conjugation in y* is the Wigner operator for time reversal. If y is a retarded wave, then y* is an advanced wave.
If y = A ei(kr-wt) then y* = A ei(-kr+wt)
(retarded) (advanced)

17. Lessons from Classical E&M

18. Maxwell’s Electromagnetic Wave Equation
Ñ2 Fi = 1/c2 ¶2Fi /¶t2
This is a 2nd order differential equation, which has two time solutions, retarded and advanced.

19. Maxwell’s Electromagnetic Wave Equation
Ñ2 Fi = 1/c2 ¶2Fi /¶t2
This is a 2nd 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
Ñ2 Fi = 1/c2 ¶2Fi /¶t2
This is a 2nd 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
Interpret Maxwell’s wave equation as a relativistic quantum wave equation (for mrest = 0).

27. The Logic of the Transactional Interpretation
Interpret Maxwell’s wave equation as a relativistic quantum wave equation (for mrest = 0).
Interpret the relativistic Klein-Gordon and Dirac equations (for mrest > 0)

28. The Logic of the Transactional Interpretation
Interpret Maxwell’s wave equation as a relativistic quantum wave equation (for mrest = 0).
Interpret the relativistic Klein-Gordon and Dirac equations (for mrest > 0)
Interpret the Schrödinger equation as a non-relativistic reduction of the K-G and Dirac equations (for mrest > 0).

29. The Quantum Transactional Model
Step 1: The emitter sends out an “offer wave” Y.

30. The Quantum Transactional Model
Step 1: The emitter sends out an “offer wave” Y.
Step 2: The absorber responds with a “confirmation wave” Y*.

31. The Quantum Transactional Model
Step 1: The emitter sends out an “offer wave” Y.
Step 2: The absorber responds with a “confirmation wave” Y*.
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 advanced- wave 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 advanced- wave E-field at the emitter.
Translating this to quantum mechanical terms, the “echo” that the emitter receives from each potential absorber is yy*, leading to the Born Probability Law. y
yy*

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. 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.

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? (Y = 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? (Y = 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 T1 and T2.

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 T1 and T2.
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 T1 or T2 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 (T1, T2) 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(qrel) = Cos2qrel]

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(qrel) = Cos2qrel]
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(qrel) = Cos2qrel]
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 advanced-retarded 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 advanced- retarded 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 D1 telegraph information as the measurement outcome at D2?

61. Is FTL Communication Possible with EPR Nonlocality?
Question: Can the choice of measurements at D1 telegraph information as the measurement outcome at D2?
Answer: No! Operators for measurements D1 and D2 commute. [D1, D2]=0. Choice of measurements at D1 has no observable consequences at D2. (Eberhard’s Theorem)

62. Is FTL Communication Possible with EPR Nonlocality?
Question: Can the choice of measurements at D1 telegraph information as the measurement outcome at D2?
Answer: No! Operators for measurements D1 and D2 commute. [D1, D2]=0. Choice of measurements at D1 has no observable consequences at D2. (Eberhard’s Theorem)
Levels of EPR Communication:
Enforce conservation laws (Yes)

63. Is FTL Communication Possible with EPR Nonlocality?
Question: Can the choice of measurements at D1 telegraph information as the measurement outcome at D2?
Answer: No! Operators for measurements D1 and D2 commute. [D1, D2]=0. Choice of measurements at D1 has no observable consequences at D2. (Eberhard’s Theorem)
Levels of EPR Communication:
Enforce conservation laws (Yes)
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, wave-particle 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 S1 into two paths, A and B, having equal lengths, and recombines the beams at S2. All the light goes to detector D1 because the beams interfere destructively at detector D2.

72. The Mach-Zender Interferometer
A Mach-Zender intereferometer splits a light beam at S1 into two paths, A and B, having equal lengths, and recombines the beams at S2. All the light goes to detector D1 because the beams interfere destructively at detector D2.
D1: L|S1r|Ar|S2t|D1 and L|S1t|Br|S2r|D1 => in phase

73. The Mach-Zender Interferometer
A Mach-Zender intereferometer splits a light beam at S1 into two paths, A and B, having equal lengths, and recombines the beams at S2. All the light goes to detector D1 because the beams interfere destructively at detector D2.
D1: L|S1r|Ar|S2t|D1 and L|S1t|Br|S2r|D1 => in phase
D2: L|S1r|Ar|S2r|D2 and L|S1t|Br|S2t|D2 => 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 D1 and D2.

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 D1 and D2.
This is because there is now no interference, and splitter S2 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 D1 and D2.
This is because there is now no interference, and splitter S2 divides the incident light equally between the two detector paths.
Therefore, detection of a photon at D2 (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 D1. If it is a live bomb, ½ of the time the bomb will explode, ¼ of the time it will go to D1 and ¼ of the time to D2.

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 D1. If it is a live bomb, ½ of the time the bomb will explode, ¼ of the time it will go to D1 and ¼ of the time to D2.
Therefore, on each D1 signal, send in another photon. On a D2 signal, stop, you have a live bomb! After 10 or so D1 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 D1.

84. Transactions for No Object
There are two allowed paths between the light source L and the detector D1. If the paths have equal lengths, the offer waves y to D1 will interfere constructively, while the offer y waves to D2 interfere destructively and cancel.

85. Transactions for No Object
There are two allowed paths between the light source L and the detector D1. If the paths have equal lengths, the offer waves y to D1 will interfere constructively, while the offer y waves to D2 interfere destructively and cancel. The confirmation waves y* 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 S2 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 D1 and D2 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 DH will be detected with a probability PD = Cos2N(p/2N).
The photon will interact with the object with a probability PR = 1 - PD = 1 - Cos2N(p/2N). When N is large, PD » 1 - (p/2)2/N and PR » (p/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 p/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-free-measurements.
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 counter-intuitive quantum optics results.