Derivation of Born’s rule in the Many-Worlds interpretation using self-location uncertainty.

The Many-Worlds Interpretation is a deterministic formulation of quantum mechanics, which focuses on the wave nature and thus unitary transformations. Being deterministic in nature, defining probabilities in it, and deriving the Born rule has been a challenge. Sean Carroll in his work uses self-locating uncertainty- the indeterminacy of the world or ‘branch’ one is located in and the Epistemic separability principle- a given result is independent of the non-interacting and remote environment. In this work, the step-by-step process of interactions with the Apparatus, the Observer, and decoherence interactions with the environment leading to the Born rule is explained transparently and concisely. The key aspect of Carroll’s work involves the observer copying oneself into as many copies as the number of worlds before the branching is complete. This is to facilitate the self-location perspective. In addition, the work also explores the cosmological consequences of the Many-World’s interpretation like the Cosmological measure problem. We also take a brief look at some of the criticisms against Carroll’s arguments by Adrian Kent.

Transcript

1. 1/22 (Born Rule from self locating uncertainty in Everettian Quantum Mechanics1) Vijay Shankar A, vijanebula@gmail.com January 31, 2024 1Sean M Carroll and Charles T Sebens. “Many worlds, the born rule, and self-locating uncertainty”. In: Quantum theory: A two-time success story: Yakir Aharonov Festschrift. Springer. 2014, pp. 157–169. Vijay Shankar A January 31, 2024 1 / 22
2. 2/22 Born Rule Probability of obtaining an Eigenstate given that a quantum system is prepared in a state (a superposition). |Ψ⟩ = p1|ψ1⟩ + p2|ψ2⟩ (1) p2 1 = |⟨ψ1|Ψ⟩|2 (2) Probabilities are inherent in nature! Vijay Shankar A January 31, 2024 2 / 22
3. 3/22 Many-Worlds Interpretation Can we say that things are made up of probabilities? ”God does not play Dice” - Albert Einstein Many-Worlds Interpretation to the Rescue? ”All the possibilities are equally true - they exist as distinct branches of reality” -Hugh Everett Then how do you explain probabilities? Vijay Shankar A January 31, 2024 3 / 22
4. 4/22 The Many Worlds Interpretation-Carroll2 A deterministic theory that focuses on branching of realities. Observer(O)-Apparatus(A)-Environment(ω) |Ψ⟩ = |O0⟩ (|↑⟩ + |↓⟩) |A0⟩|ω0⟩ Branching begins.. → |O0⟩ (|↑⟩|A↑⟩ + |↓⟩|A↓⟩) |ω0⟩ Decoherence → |O0⟩ (|↑⟩|A↑⟩|ω↑⟩ + |↓⟩|A↓⟩|ω↓⟩) Branching complete → |O↑⟩|↑⟩|A↑⟩|ω↑⟩ + |O↓⟩|↓⟩|A↓⟩|ω↓⟩ 2Carroll and Sebens, “Many worlds, the born rule, and self-locating uncertainty”. Vijay Shankar A January 31, 2024 4 / 22
5. 5/22 What happens with the Observer?3 Imagine Alice is standing in front of two doors and has to choose one. The flip of a coin helps her decide. Imagine someone gives Alice a sleeping pill and she finds herself in one of the rooms. She can’t know where she is - this is called self-location Uncertainty. Figure: location choice and uncertainty 3Kelvin J McQueen and Lev Vaidman. “In defence of the self-location uncertainty account of probability in the many-worlds interpretation”. In: Studies in History and Vijay Shankar A January 31, 2024 5 / 22
6. 6/22 Self-Locating Uncertainty But before the last step, the observer copies into two: = |O0⟩|↑⟩|A↑⟩|ω↑⟩ + |O0⟩|↓⟩|A↓⟩|ω↓⟩ The observer can’t know in which branch one joins before the measurement - a short Decoherence interval prevents direct observation. What is the probability of finding oneself in a particular world?. Vijay Shankar A January 31, 2024 6 / 22
7. 7/22 Epistemic Seperability Principle All possible unrelated states that arise due to Unitary Evolutions by the environment must have equal branching probabilities. It is useful to convert unequal probability amplitudes to the sum of equal amplitudes. P(O measures s|Ψ) = P(O measures s|UE [Ψ]). UE is Unitary evolution by the Environment. Vijay Shankar A January 31, 2024 7 / 22
8. 8/22 Observer(O)-System(S)-Environment(E) Hilbert space decomposition H = HO ⊗ HS ⊗ HE Unitary Evolution UE = µ |˜ ωµ⟩⟨ωµ|, Useful decomposition |˜ ω1⟩ = |H⟩ ⊗ |Ω1⟩, |˜ ω2⟩ = |T⟩ ⊗ |Ω2⟩, Vijay Shankar A January 31, 2024 8 / 22
9. 9/22 Two evolutions A simple two-state system |Ψ⟩ = |O⟩|↑⟩|ω1⟩ + |O⟩|↓⟩|ω2⟩. Different environmental Unitary operators U(1) E = µ |˜ ωµ⟩⟨ωµ|. . U(2) E = |˜ ω2⟩⟨ω1| + |˜ ω1⟩⟨ω2| + µ>2 |˜ ωµ⟩⟨ωµ|. Vijay Shankar A January 31, 2024 9 / 22
10. 10/22 Two branches Creation of ESP states - direct |Ψ1⟩ ≡ 1OS ⊗ U(1) E |Ψ⟩ = |O⟩|↑⟩|H⟩|Ω1⟩ + |O⟩|↓⟩|T⟩|Ω2⟩ Creation of ESP states - cross |Ψ2⟩ ≡ 1OS ⊗ U(2) E |Ψ⟩ = |O⟩|↑⟩|T⟩|Ω2⟩ + |O⟩|↓⟩|H⟩|Ω1⟩. Notice the uniqueness..will be visible better.. Vijay Shankar A January 31, 2024 10 / 22
11. 11/22 Equal probabilities One has to choose the math(basis states) to ensure that the Epistemic separability principle is satisfied, as result: P(↑|Ψ) = P(↑|Ψ1) = P(↑|Ψ2), P(↓|Ψ) = P(↓|Ψ1) = P(↓|Ψ2). P(H|Ψ1) = P(H|Ψ2), P(T|Ψ1) = P(T|Ψ2). Vijay Shankar A January 31, 2024 11 / 22
12. 12/22 Equal probabilties Within the branch, different basis probabilities are also equal. P(↑|Ψ1) = P(H|Ψ1), P(↓|Ψ2) = P(H|Ψ2). Finally, ESP leads to Born Rule: P(↑|Ψ) = P(↓|Ψ) = 1/2. Interchange observer and the system O Ω ↑ T H ↓ O Ω ↑ T H ↓ Ψ | 〉 1 Ψ | 〉 2 i i Figure: Observer+System credits: Sean Carroll Vijay Shankar A January 31, 2024 12 / 22
13. 13/22 Three branches Unequal amplitudes can give a better perspective |Ψ⟩ = |O⟩|↑⟩|ω1⟩ + √ 2|O⟩|↓⟩|ω2⟩. Additional Unitary transform required U(0) E = |ω1⟩⟨ω1| + 1 √ 2 |ω2⟩ + 1 √ 2 |ω3⟩ ⟨ω2| + 1 √ 2 |ω2⟩ − 1 √ 2 |ω3⟩ ⟨ω3| + µ>3 |ωµ⟩⟨ωµ| For equal amplitudes |Ψ0⟩ = |O⟩|↑⟩|ω1⟩ + |O⟩|↓⟩|ω2⟩ + |O⟩|↓⟩|ω3⟩. Vijay Shankar A January 31, 2024 13 / 22
14. 14/22 Environment decomposition Can you notice a specific pattern of uniqueness? |˜ ω1⟩ = |+⟩ ⊗ |♡⟩ ⊗ |Ω1⟩, |˜ ω2⟩ = |−⟩ ⊗ |♢⟩ ⊗ |Ω2⟩ |˜ ω3⟩ = |0⟩ ⊗ |♠⟩ ⊗ |Ω3⟩, |˜ ω4⟩ = |−⟩ ⊗ |♠⟩ ⊗ |Ω4⟩ |˜ ω5⟩ = |0⟩ ⊗ |♡⟩ ⊗ |Ω5⟩ |˜ ω6⟩ = |+⟩ ⊗ |♢⟩ ⊗ |Ω6⟩ For example, ω1 to ω3 does not have two spades etc.. Vijay Shankar A January 31, 2024 14 / 22
15. 15/22 Environment Unitary Decomposition Similar to the two branches casw we have: the direct: U(1) E = µ |˜ ωµ⟩⟨ωµ|, and the cross: U(2) E = 3 µ=1 |˜ ωµ+3⟩⟨ωµ| + 6 µ=4 |˜ ωµ−3⟩⟨ωµ| + µ>6 |˜ ωµ⟩⟨ωµ|. The result is similar to the previous figure, notice additional terms, why are they necessary? Vijay Shankar A January 31, 2024 15 / 22
16. 16/22 Resultant branches Similar to the two-states case, we have direct and cross-states. |Ψ1⟩ = |O⟩|↑⟩|+⟩|♡⟩|Ω1⟩ + |O⟩|↓⟩|−⟩|♢⟩|Ω2⟩ + |O⟩|↓⟩|0⟩|♠⟩|Ω3⟩ |Ψ2⟩ = |O⟩|↑⟩|−⟩|♠⟩|Ω4⟩ + |O⟩|↓⟩|0⟩|♡⟩|Ω5⟩ + |O⟩|↓⟩|+⟩|♢⟩|Ω6⟩. Notice the significance of the uniqueness we maintained earlier that will ensure these states. Vijay Shankar A January 31, 2024 16 / 22
17. 17/22 Equal Probabilities Any of the basis eigenstates can be measured and their probabilities are equal P(↑|Ψ2) = P(−|Ψ2) = P(♠|Ψ2). Different Eigenstates in both the cross and direct states are also equal P(↑|Ψ1) = P(↑|Ψ2), P(−|Ψ1) = P(−|Ψ2), P(♠|Ψ1) = P(♠|Ψ2). Finally, We get the Born rule: P(↑|Ψ) = 1 2 P(↓|Ψ) = 1 3 . Vijay Shankar A January 31, 2024 17 / 22
18. 18/22 Cosmological measure problem Vijay Shankar A January 31, 2024 18 / 22
19. 19/22 Quantum-Classical ”The probability of the Observer finding oneself as one of the identical observers is the same as the Observer finding oneself in one of the branches” -Sean Carroll Probability of being one of the identical observers, given a Universe P(Oi |U) = wi j wj . here wi = |ψi |2, and total is not equal to 1! Extended Born Rule: |σi ⟩ = αi |↑⟩ + βi |↓⟩ P(↑|U) = i |αi |2P(Oi |U). Vijay Shankar A January 31, 2024 19 / 22
20. 20/22 Criticisms4 Splitting into two observers pre-measurement is not necessary and not clearly defined. Decomposition of the Environment basis states is arbitrary The definition of the Environment and Observer is not clear. Observer splitting into two copies is unnecessary. Recoherence and time evolution of the observer are not compatible. The entire procedure lacks rigor - principles, postulates, and rules are missing. 4Adrian Kent. “Does it make sense to speak of self-locating uncertainty in the universal wave function? Remarks on Sebens and Carroll”. In: Foundations of Physics 45 (2015), pp. 211–217. Vijay Shankar A January 31, 2024 20 / 22
21. 21/22 Conclusion A deterministic derivation of Born’s rule. Does not consider the case of infinite observers - has a strong dependence on Cosmology. Emphasises a strong role for the environment. entirely dependent on Hilbert space mathematics. Vijay Shankar A January 31, 2024 21 / 22
22. 22/22 References Carroll, Sean M and Charles T Sebens. “Many worlds, the born rule, and self-locating uncertainty”. In: Quantum theory: A two-time success story: Yakir Aharonov Festschrift. Springer. 2014, pp. 157–169. Kent, Adrian. “Does it make sense to speak of self-locating uncertainty in the universal wave function? Remarks on Sebens and Carroll”. In: Foundations of Physics 45 (2015), pp. 211–217. McQueen, Kelvin J and Lev Vaidman. “In defence of the self-location uncertainty account of probability in the many-worlds interpretation”. In: Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 66 (2019), pp. 14–23. Vijay Shankar A January 31, 2024 22 / 22