Entropy & matter T violation anomaly Quantum theory of time Sources of T violation Experimental test The quantum theory of time: from formalism to experimental test

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Toshio Croucher Joan Vaccaro leaves on Centre for Quantum Dynamics ground on Griffith University this side Brisbane Australia of tree

1 / 20 Entropy & matter T violation anomaly Quantum theory of time Sources of T violation Experimental test Vaccaro, Proc. R. Soc. A 472, 20150670 (2016)

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2 / 20 Entropy & & matter matter T violation anomaly Quantum theory of time Sources of T violation Experimental test Entropy & matter anomalies

Recall Loschmidt’s paradox:

Evolving “upwards” in time: − entropy increases. t y Consider a gas with N, T, P, & V known, x in a container that has a leak.

Evolving “downwards” in time: − entropy increases.

Symmetry is due to time symmetric dynamical laws.

We take entropy increase as evidence of the “direction of time evolution” or “direction of time” for short. 3 / 20 Entropy & & matter matter T violation anomaly Quantum theory of time Sources of T violation Experimental test Empirical evidence: all space and time has a consistent direction of entropy increase:

Distant galaxy

effects

direction direction of of entropy time increase evolution Earth

causes

4 / 20 Entropy & & matter matter T violation anomaly Quantum theory of time Sources of T violation Experimental test From Loschmidt’s paradox, expect symmetry in time: high entropy? low entropy high entropy

− t time evolution? 0 time evolution t

Similarly, expect symmetry in matter and antimatter:

antimatter? radiation matter − t 0 t There is no external reference for the direction of time. So the universe must be time symmetric overall.

These are the entropy and matter-antimatter anomalies. Proposed resolutions: entropy 2004 Carrol & Chen hep-th/0410270 high low high 2011 Vaccaro Found Phys 41, 1596 − t 0 t 2014 Barbour, Koslowski & Mercati antimatter radiation matter Phys Rev Lett 113 181101 (2014) − t 0 t 2018 Boyle, Finn & Turok Phys Rev Lett 121 251301 (2018) 5 / 20 Entropy & matter TT violation violation anomalyanomaly Quantum theory of time Sources of T violation Experimental test T violation anomaly

Recall the discrete symmetries: z x y t t C P

0 0 T 𝑝𝑝− 𝑛𝑛 𝐾𝐾0 𝑝𝑝̅+ 𝑛𝑛� 𝐾𝐾�0 x y z � 𝑒𝑒 𝜈𝜈 𝐵𝐵 𝑒𝑒 𝜈𝜈̅ 𝐵𝐵 P parity inversion C charge conjugation x ↔ −x T time reversal y ↔ −y particle ↔ antiparticle z ↔ −z t ↔ −t History: P violation Lee & Yang 1956 β− decay in 60Co P CP violation Cronin & Fitch 1964 Wu’s experiment 0 K decay ( + 0) + +− 𝐾𝐾𝐿𝐿 →≠ 𝜋𝜋CP=𝜋𝜋 1 −0 0+ CP=𝐾𝐾−1 CP=𝜖𝜖𝜖𝜖+ 1 T violation BarBar, SLAC 2012 0 B decay T

0 0 𝐵𝐵� → 𝐵𝐵− ≠ 𝐵𝐵− → 𝐵𝐵� 6 / 20 Entropy & matter TT violation violation anomalyanomaly Quantum theory of time Sources of T violation Experimental test T violation in B0 system weak interaction + Experiment: BarBar Collaboration, SLAC, Stanford University 𝐵𝐵+ 𝑖𝑖𝐵𝐵− J. P. Lees et al., Phys. Rev. Lett. 109, 211801 (2012). Detailed description: 0 0 Bernabeu et al, JETP Lett. 2012, 64 (2012). 𝐵𝐵decay decay𝐵𝐵� + − Empirical 𝐵𝐵 − 𝑖𝑖𝐵𝐵 Hamiltonian

0 0 𝐵𝐵 𝐵𝐵� 0 * 𝐵𝐵

time 0 reversal𝐵𝐵� operation T violation: different ∆τ

Time-reversed Hamiltonian𝕋𝕋� 0 *0 𝐵𝐵 𝐵𝐵� 0 e e −1 𝐵𝐵 −𝑖𝑖𝐻𝐻�Δ𝜏𝜏 +e𝑖𝑖𝕋𝕋�𝐻𝐻�𝕋𝕋� Δ𝜏𝜏 0 0 𝑡𝑡 0 𝑖𝑖𝐻𝐻�Δ𝜏𝜏 𝑡𝑡 7 / 20 𝐵𝐵� 𝐵𝐵� 𝐵𝐵− 𝐵𝐵� 𝐵𝐵− Entropy & matter TT violation violation anomalyanomaly Quantum theory of time Sources of T violation Experimental test

T violation implies 2 Hamiltonians and 2 dynamics – one for each direction of time. Physical theory needs to account for both Hamiltonians. Conventional physical theories have a single version of the Hamiltonian and the time evolution it describes (into the future, by default). Phil. Tras. A 376, 20170316 (2018). This is the T violation anomaly Proposed resolution: The quantum theory of time arXiv: 1605.01965

Invited Chapter in book: S. Wuppuluri, G. Ghirardi Eds, "Space, Time and the Limits of Human Understanding" (Springer International, 2016) 185-201 8 / 20 Entropy & matter T violation anomaly QuantumQuantum theorytheory ofof timetime Sources of T violation Experimental test

Quantum theory of time worldline of particle Space-time background itself: 𝑡𝑡 𝑡𝑡 - space and time are essentially interconvertible = 2 = 2 2 2 2 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 2 − 𝑑𝑑𝑑𝑑 2 − 𝑑𝑑𝑑𝑑 2 − 𝑑𝑑𝑑𝑑 2 𝑥𝑥 Things on space-time background:𝑑𝑑𝑡𝑡 − 𝑑𝑑𝑥𝑥 − 𝑑𝑑𝑦𝑦 − 𝑑𝑑𝑧𝑧 𝑥𝑥 Dynamics (relativistic, quantum, classical): - defines translations over time but not space. treat as a Conservation laws (energy, momentum, lepton #,...): phenomenological - apply over time but not over space.

e.g. Quantum Mechanics: wave function over space wave function over time ( ) ( ) why not this? 𝜓𝜓 𝑥𝑥 𝜙𝜙 𝑡𝑡 mass not conserved

9 / 20 𝑥𝑥 𝑡𝑡 Entropy & matter T violation anomaly QuantumQuantum theorytheory ofof timetime Sources of T violation Experimental test

The idea is this:

fundamental level phenomenological ( , ) ( , )

𝜓𝜓 𝑥𝑥 𝑡𝑡 phenomenological 𝜓𝜓 𝑥𝑥 𝑡𝑡 𝑡𝑡 𝑡𝑡 details

no dynamics dynamics emerge 𝑥𝑥 𝑥𝑥 Need to look at phenomenology of generators of translations in space and time: • the Hamiltonian generates translations in time C,P,T discrete e e symmetries are −1 � �� � t violated by the • +𝑖𝑖𝕋𝕋𝐻𝐻𝕋𝕋 𝑡𝑡 −𝑖𝑖𝐻𝐻𝑡𝑡 the momentum operator generates translations in space Hamiltonian only! e e x +𝑖𝑖𝑃𝑃�𝑥𝑥 −𝑖𝑖𝑃𝑃�𝑥𝑥

10 / 20 Entropy & matter T violation anomaly QuantumQuantum theorytheory ofof timetime Sources of T violation Experimental test

Four principles Vaccaro, Phil. Trans. Roy. Soc. A 376, 20170316 (2018). 1. States have same construction in time and space Retain the symmetry of the space time background

2. Time evolution is directional , e “forwards” , e−𝑖𝑖𝐻𝐻�𝐹𝐹𝛿𝛿𝛿𝛿 “backwards” �𝐹𝐹 � 𝐻𝐻 ≡ 𝐻𝐻 −1 𝑖𝑖𝐻𝐻�𝐵𝐵𝛿𝛿𝛿𝛿 𝐻𝐻�𝐵𝐵 ≡ 𝕋𝕋�𝐻𝐻�𝕋𝕋� time evolution physical dynamics ≡ causality

“backwards” e e “forwards” 𝑖𝑖𝐻𝐻�𝐵𝐵𝛿𝛿𝛿𝛿 −𝑖𝑖𝐻𝐻�𝐹𝐹𝛿𝛿𝛿𝛿 e e −mathematical𝑖𝑖𝐻𝐻�𝐵𝐵𝛿𝛿𝛿𝛿 inverses𝑖𝑖𝐻𝐻�𝐹𝐹𝛿𝛿𝛿𝛿 ≡ rewind 𝑡𝑡 11 / 20 Entropy & matter T violation anomaly QuantumQuantum theorytheory ofof timetime Sources of T violation Experimental test 3. Fundamental resolution limit E.g. Planck scale, or other limit (the actual limit is not important)

4. A state is represented by a quantum path with step sizes below resolutionT symmetry limit = Space: Time: T symmetry = = + 𝐵𝐵 𝐹𝐹 = 𝑁𝑁 𝐻𝐻� 𝐻𝐻� 𝐵𝐵 𝐹𝐹 𝑖𝑖𝑃𝑃� 𝛿𝛿𝑥𝑥 2 −𝑖𝑖𝑃𝑃� 𝛿𝛿𝑥𝑥 e ee 𝐻𝐻e� 𝐻𝐻� 𝐻𝐻� 𝑒𝑒 𝑒𝑒 𝑖𝑖𝑃𝑃�𝛿𝛿𝛿𝛿 −𝑖𝑖𝑃𝑃�𝛿𝛿𝛿𝛿 t e e 𝑁𝑁 = +𝑖𝑖𝐻𝐻�𝑡𝑡 −𝑖𝑖𝐻𝐻�𝑡𝑡 𝜓𝜓 𝜎𝜎𝑋𝑋 | � � x +𝑖𝑖𝑃𝑃𝑥𝑥 −𝑖𝑖𝑃𝑃𝑥𝑥 𝛿𝛿𝛿𝛿 𝑁𝑁 | 𝜓𝜓⟩

𝜓𝜓⟩ 𝜎𝜎 𝛿𝛿𝛿𝛿 𝜎𝜎 + random 𝑡𝑡paths = 𝑁𝑁 random 𝑥𝑥paths 𝑖𝑖𝐻𝐻� 𝛿𝛿𝛿𝛿 2 −𝑖𝑖𝐻𝐻� 𝛿𝛿𝛿𝛿 step size 𝑁𝑁 = 𝑒𝑒 𝑒𝑒 e 2 no dynamics𝜓𝜓 𝜎𝜎𝑇𝑇 𝑥𝑥 𝑥𝑥 step size − 2 𝛿𝛿𝛿𝛿 𝛿𝛿𝛿𝛿 𝑁𝑁 4𝜎𝜎 mass not ≈conserved,∫ 𝑑𝑑𝑑𝑑 no𝑥𝑥 X dynamics 𝛿𝛿𝛿𝛿 12 / 20 Entropy & matter T violation anomaly QuantumQuantum theorytheory ofof timetime Sources of T violation Experimental test T violation , = Get interference between different paths to the same point in time. 𝐻𝐻�𝐵𝐵 𝐻𝐻�𝐹𝐹 𝑖𝑖𝑖𝑖𝕀𝕀� | = + = 𝑁𝑁 𝜓𝜓⟩ 𝑝𝑝 2𝜋𝜋 𝑛𝑛 𝑖𝑖𝐻𝐻�𝐵𝐵𝛿𝛿𝛿𝛿 2 −𝑖𝑖𝐻𝐻�𝐹𝐹T𝛿𝛿𝛿𝛿 violation 𝑡𝑡 𝐻𝐻�𝐵𝐵 𝐻𝐻�𝐹𝐹 𝜎𝜎𝜎𝜎 𝑁𝑁 = 𝑒𝑒 𝑒𝑒 𝜓𝜓 𝜎𝜎𝑇𝑇 𝐻𝐻�𝐵𝐵 ≠ 𝐻𝐻�𝐹𝐹 𝛿𝛿𝛿𝛿 𝑁𝑁 ⋯ ⋯ ⋯ ⋯ 𝑡𝑡 e e −𝑡𝑡𝑝𝑝 𝑡𝑡𝑝𝑝 | | 𝑖𝑖𝐻𝐻�𝐵𝐵𝛿𝛿𝑡𝑡 −𝑖𝑖𝐻𝐻�𝐹𝐹𝛿𝛿𝑡𝑡 𝜓𝜓−⟩ destructive𝜓𝜓+⟩ interference

Schrodinger equations | 𝑑𝑑 𝜓𝜓+ 𝑡𝑡 ≈ −𝑖𝑖𝐻𝐻�𝐹𝐹 𝜓𝜓+ 𝑡𝑡 〉 𝑑𝑑𝑑𝑑 | dynamics𝑑𝑑 destructive interference 𝜓𝜓− 𝑡𝑡 ≈ −𝑖𝑖𝐻𝐻�𝐵𝐵 𝜓𝜓− 𝑡𝑡 〉 13 / 20 𝑑𝑑𝑑𝑑 𝑡𝑡 Entropy & matter T violation anomaly Quantum theory of time SourcesSources ofof T violation Experimental test Sources of T violation Croucher & Vaccaro, in preparation (2019) weak interaction , , Mesons 0 0 Particle0 and0 antiparticle0 0 mixing: = 𝐵𝐵 𝐵𝐵� 0 𝐾𝐾 𝐾𝐾� 𝐵𝐵 𝐵𝐵� 1 * 𝐵𝐵 Many particles 0 0 �𝐹𝐹 0  group contraction decay decay� 𝐻𝐻T 𝐵𝐵 𝐵𝐵 𝐵𝐵�  harmonic oscillator: 0 *0 Inonu & Wigner P Natl Acad Sci USA 39 510 (1953) ̂− 𝐵𝐵 𝐵𝐵� Arecchi et al Phys Rev A 6 2211 (1972) 𝐽𝐽 → 𝑎𝑎�† = 0 ̂+ 𝐵𝐵 ( ) 𝐽𝐽 → 𝑎𝑎� 1 = , = Branco, Silva 𝐵𝐵& Lavoura CP Violation0 (1999) 𝑧𝑧 𝐻𝐻� 𝑛𝑛 𝐽𝐽̂ 𝐵𝐵� ̂𝜇𝜇 𝜇𝜇 𝐽𝐽 � 𝜎𝜎� 𝑥𝑥 𝑦𝑦 𝑧𝑧 𝑛𝑛 𝚥𝚥̂ 𝚥𝚥̂ 𝑖𝑖𝐽𝐽̂ = + + 𝑁𝑁 = † + + ∗ † 𝐻𝐻�𝐹𝐹 𝜔𝜔𝑎𝑎� 𝑎𝑎� 𝑁𝑁𝜶𝜶𝑎𝑎� 𝑁𝑁𝜶𝜶 𝑎𝑎� 𝑁𝑁 † ∗ † , = �𝐹𝐹 ̂𝑥𝑥 ̂𝑦𝑦 𝐻𝐻 𝜔𝜔𝑎𝑎� 𝑎𝑎� 𝑁𝑁𝜶𝜶 𝑎𝑎� 𝑁𝑁𝜶𝜶𝑎𝑎� 𝐽𝐽 𝐽𝐽 , � 𝑁𝑁 𝑁𝑁 𝟐𝟐 𝑥𝑥� 𝑝𝑝̂ 𝑖𝑖𝕀𝕀 𝐹𝐹 𝐵𝐵 𝐻𝐻� 𝐻𝐻� 𝑁𝑁≫𝜔𝜔 ≈ 𝑖𝑖 𝑁𝑁 𝜶𝜶 𝕀𝕀� 14 / 20 𝑥𝑥� 𝑝𝑝̂ Entropy & matter T violation anomaly Quantum theory of time SourcesSources ofof T violation Experimental test Neutrinos Nunokawa Prog. Part. Nucl. Phys. 60 338 (2008) In rotated Flavour basis: Suspected of violating CP and T symmetries 2D subspace is just like mesons Flavour mixing: ( ) = * = 1 𝑒𝑒 𝑒𝑒 𝐹𝐹 , 𝜈𝜈 1 𝜈𝜈 𝐻𝐻� * 𝑑𝑑 𝜇𝜇 𝜇𝜇 𝑁𝑁 𝑁𝑁 𝟐𝟐 𝜈𝜈 � 𝜈𝜈 ( ) 𝐹𝐹 𝐵𝐵 𝐻𝐻 𝐻𝐻� 𝐻𝐻� 𝑁𝑁≫1 ≈ 𝑖𝑖 𝑁𝑁 𝜶𝜶 𝕀𝕀� 𝑑𝑑𝑑𝑑 𝜈𝜈𝜏𝜏 𝜈𝜈𝜏𝜏 = 1 𝐵𝐵 Higgs-like scalar field Branco et𝐻𝐻� al., Phys Rep., 516, 1 (2012) Spontaneous symmetry breaking: CP & T violation arises in 2 or more doublet models Explicit symmetry breaking in scalar field: = + + + + 2 2 2 2 � zero � = � �+ + ℋ momentum𝜋𝜋� ∇𝜙𝜙 𝑚𝑚 𝜙𝜙 𝒂𝒂𝜙𝜙 𝒃𝒃𝜋𝜋� , † ∗ † mode = + + 𝐻𝐻�0𝐹𝐹 𝑚𝑚𝑎𝑎�0𝑎𝑎�0 𝜶𝜶𝑎𝑎�0 𝜶𝜶 𝑎𝑎�0 𝟐𝟐 † † �0𝐹𝐹 �0𝐵𝐵 𝛼𝛼 ≫𝑚𝑚 � ∗ 𝐻𝐻 𝐻𝐻 ≈ 𝑖𝑖 𝜶𝜶 𝕀𝕀 15 / 20 𝐻𝐻�0𝐵𝐵 𝑚𝑚𝑎𝑎�0𝑎𝑎�0 𝜶𝜶 𝑎𝑎�0 𝜶𝜶𝑎𝑎�0 Entropy & matter T violation anomaly Quantum theory of time Sources of T violation ExperimentalExperimental testtest Experimental test Vaccaro, in preparation (2019)

Spatial distribution – T symmetry case

e + e 𝑁𝑁 𝑖𝑖𝑯𝑯�𝜎𝜎 𝑖𝑖𝑯𝑯�𝜎𝜎 𝑡𝑡 = + + e 2 2 = lim − 𝑯𝑯� 𝜎𝜎 𝑁𝑁 2 𝑁𝑁 2 2 − � �𝐴𝐴 �𝐵𝐵 �𝐶𝐶 2 𝑯𝑯 𝐻𝐻 𝐻𝐻 𝐻𝐻 𝑁𝑁→∞ 𝑨𝑨 𝑩𝑩 𝑪𝑪 𝑥𝑥 + + correlated fluctuations 2 2 2 𝐻𝐻�𝐴𝐴 𝐻𝐻�𝐵𝐵( 𝐻𝐻�𝐶𝐶 ) e 2 2 2 2 = e 2 2e 2 2e 2 2 𝐻𝐻�𝑨𝑨 +𝐻𝐻�𝑩𝑩 +𝐻𝐻�𝐶𝐶 𝜎𝜎 𝐻𝐻�𝑨𝑨 𝜎𝜎 𝐻𝐻�𝑩𝑩 𝜎𝜎 𝐻𝐻�𝐶𝐶 𝜎𝜎 − − − − 2 2 2 2 𝑡𝑡 e + e 𝑁𝑁 e + e 𝑁𝑁 e + e 𝑁𝑁 𝑖𝑖𝑯𝑯� 𝐴𝐴𝜎𝜎 𝑖𝑖𝑯𝑯� 𝑨𝑨𝜎𝜎 𝑖𝑖𝑯𝑯� 𝐵𝐵𝜎𝜎 𝑖𝑖𝑯𝑯� 𝑩𝑩𝜎𝜎 𝑖𝑖𝑯𝑯� 𝐶𝐶𝜎𝜎 𝑖𝑖𝑯𝑯� 𝑪𝑪𝜎𝜎 − − − 𝑁𝑁 2 𝑁𝑁 𝑁𝑁 2 𝑁𝑁 𝑁𝑁 2 𝑁𝑁 𝑨𝑨 𝑩𝑩 𝑪𝑪 independent fluctuations𝑥𝑥 , but no interactions 16 / 20 Entropy & matter T violation anomaly Quantum theory of time Sources of T violation ExperimentalExperimental testtest

1 1 1 1 + + = + + + + + 3 3 3 3 2 2 2 2 2 2 2 𝐻𝐻�𝐴𝐴 𝐻𝐻�𝐵𝐵 𝐻𝐻�𝐶𝐶 𝐻𝐻�𝐴𝐴 𝐻𝐻�𝐵𝐵 𝐻𝐻�𝐶𝐶 𝐻𝐻�𝐴𝐴 − 𝐻𝐻�𝐵𝐵 𝐻𝐻�𝐵𝐵 − 𝐻𝐻�𝐶𝐶 𝐻𝐻�𝐴𝐴 − 𝐻𝐻�𝐶𝐶

( ) ( ) ( ) ( ) ( ) ( 1 1 𝑁𝑁 1 1 𝑁𝑁 1 1 e𝑖𝑖 𝑯𝑯� 𝐴𝐴+ ⋯ 𝜎𝜎 + e 𝑖𝑖 𝑯𝑯� 𝐴𝐴+ ⋯ 𝜎𝜎 e𝑖𝑖 𝑯𝑯� 𝐴𝐴−𝑯𝑯� 𝑩𝑩 𝜎𝜎 + e 𝑖𝑖 𝑯𝑯� 𝐴𝐴−𝑯𝑯� 𝑩𝑩 𝜎𝜎 e𝑖𝑖 𝑯𝑯� 𝐵𝐵−𝑯𝑯� 𝑪𝑪 𝜎𝜎 + e 𝑖𝑖 𝑯𝑯� 𝐵𝐵− 3 − 3 3 − 3 3 − 3 𝑁𝑁 2 𝑁𝑁 𝑁𝑁 2 𝑁𝑁 𝑁𝑁 2 𝑁𝑁

allows interactions allows independent fluctuations

One last thing…. 𝑡𝑡 The value of needs to adjust locally to maintain minimum uncertainty in 𝑨𝑨 𝑩𝑩 𝑪𝑪 independent fluctuations,𝑥𝑥 energy, thus𝜎𝜎 with interactions + + + +

17 / 20 𝐴𝐴 �𝐵𝐵 �𝐶𝐶 𝐴𝐴 𝐴𝐴 𝐵𝐵 𝐵𝐵 𝐶𝐶 𝐶𝐶 𝐻𝐻� 𝐻𝐻 𝐻𝐻 𝜎𝜎 ↦ 𝐻𝐻� 𝜎𝜎 𝐻𝐻� 𝜎𝜎 𝐻𝐻� 𝜎𝜎 Entropy & matter T violation anomaly Quantum theory of time Sources of T violation ExperimentalExperimental testtest

( ) ( ) ( ) ( ) ( ) 1 1 𝑁𝑁 1 1 𝑁𝑁 1 e𝑖𝑖 𝑯𝑯� 𝐴𝐴𝜎𝜎𝐴𝐴+⋯ + e 𝑖𝑖 𝑯𝑯� 𝐴𝐴𝜎𝜎𝐴𝐴+⋯ e𝑖𝑖 𝑯𝑯� 𝐴𝐴𝜎𝜎𝐴𝐴−𝑯𝑯� 𝑩𝑩𝜎𝜎𝐵𝐵 + e 𝑖𝑖 𝑯𝑯� 𝐴𝐴𝜎𝜎𝐴𝐴−𝑯𝑯� 𝑩𝑩𝜎𝜎𝐵𝐵 e𝑖𝑖 𝑯𝑯� 𝐵𝐵𝜎𝜎𝐵𝐵−𝑯𝑯� 𝑪𝑪𝝈𝝈+𝑪𝑪 3 − 3 3 − 3 3 𝑁𝑁 2 𝑁𝑁 𝑁𝑁 2 𝑁𝑁 𝑁𝑁 2

T violation case 𝑡𝑡 𝑪𝑪 𝑨𝑨 𝑩𝑩

𝑥𝑥

𝑨𝑨 𝑩𝑩 𝑪𝑪𝑪𝑪 independent fluctuations, with interactions and local effects 18 / 20 Entropy & matter T violation anomaly Quantum theory of time Sources of T violation ExperimentalExperimental testtest Experimental test Local variation in T violation local variation in clock time = 2𝜋𝜋 𝑁𝑁 Antineutrinos 𝑐𝑐 𝑡𝑡 𝑟𝑟 = , 𝜎𝜎 1 𝑟𝑟 �𝐵𝐵 �𝐹𝐹 𝜎𝜎 1𝑖𝑖+ 𝐻𝐻 𝐻𝐻 𝑟𝑟 is the ∝ratio ofΛ the2 strengths of the T violation due to 𝑟𝑟the reactor neutrinos andΛ the background processes at 1 m density 1 generated power ≈4 GW 2 ∝ 𝑟𝑟

quantum clock dys-schronisation: ( ) = 1 where = 1 Λ 2 19 / 20 𝑡𝑡𝑐𝑐 𝑟𝑟 𝛾𝛾𝑟𝑟𝑡𝑡𝑐𝑐 𝛾𝛾𝑟𝑟 − 𝑟𝑟 Entropy & matter T violation anomaly Quantum theory of time Sources of T violation Experimental test Proc. Roy. Soc. A 472, 20150670 (2016) Conclusion arXiv: 1605.01965 (book chapter) Phil. Tras. A 376, 20170316 (2018).

• Violations of discrete symmetries P, C & T are yet to be fully appreciated • Generator of translations in time violates P, CP and T • Proposed here as the origin of dynamics and conservation laws

Rare opportunity: To formulate a well-posed question for Nature, the answer of which could necessitate a revision of how we treat Time in physics 20 / 20 Entropy & matter T violation anomaly Quantum theory of time Sources of T violation Experimental test Lorentz covariant extension QFT: Recall the Klein-Gordon equation for massive scalar field ( ) (+, , , ) = = + 2 mass shell: 𝜓𝜓 𝑥𝑥 − − − 𝜕𝜕 2 2 2 2 Feynman’s off mass2 shell theory for virtual (exchange) 𝐤𝐤particle described by ( , ) 𝜇𝜇 𝜓𝜓 𝑥𝑥 −𝑚𝑚 𝜓𝜓 𝑥𝑥 𝜔𝜔 𝑘𝑘 𝑚𝑚 𝜕𝜕𝑥𝑥 Feynman, Phys. Rev. 80, 440 (1950) App. A ( , ) = ( , ) 2 See𝜙𝜙 also𝑥𝑥 Fanchi𝜏𝜏 , Found. Phys. 23, 487 (1993) 𝜕𝜕 𝜕𝜕 where is “proper𝑖𝑖 time”𝜙𝜙 𝑥𝑥 𝜏𝜏 of the− particle.2 𝜙𝜙 𝑥𝑥 𝜏𝜏 Recover by projection: 𝜕𝜕𝜕𝜕 𝜕𝜕𝑥𝑥𝜇𝜇 τ = ∞ exp ( , ) 𝜓𝜓 𝑥𝑥 picks out one mass component (frequency) 2 𝜓𝜓 𝑥𝑥 � −𝑖𝑖𝑚𝑚 𝜏𝜏 𝜙𝜙 𝑥𝑥 𝜏𝜏 Here: , creates−∞ single particle from vacuum centred on point = ( , , , ) . T symmetry† T violation peak at𝜙𝜙 𝑥𝑥=𝜏𝜏 peaks at = , , … 𝑥𝑥 𝜏𝜏 𝑎𝑎 coarse𝑏𝑏 𝑐𝑐 graining fixes mass

𝑥𝑥0 𝜏𝜏 𝑥𝑥0 𝜏𝜏1 𝜏𝜏2 † † 0 † 𝑥𝑥 0 𝜙𝜙 𝜙𝜙 𝑥𝑥0 𝜙𝜙 𝜙𝜙 𝜙𝜙 𝜙𝜙 𝑥𝑥 𝜏𝜏2 coarse 1 graining 𝜏𝜏 𝜏𝜏

𝑎𝑎 𝑥𝑥1 𝑎𝑎 𝑥𝑥1 𝑎𝑎 𝑥𝑥1