Quantum asymmetry between time and space & the origin of dynamics Joan Vaccaro Centre for Quantum Dynamics Griffith University Brisbane Australia 1 / 17

DOI: 10.1098/rspa.2015.0670 2 / 17

The hard problem of time J. Ellis McTaggart – The Unreality of Time (Mind, 1908) ceases to exist 9:00 past 10:00 now 11:00 future 9:00 timeless 10:00 ordered 11:00 sequence exists not logical not yet existing t0 x t y entropy William James (Dilemma of Determinism 1884) likened a deterministic universe as rigid as an iron block - now called the Block Universe. Aushalom Elitur Problem Why does time have a direction? How can it be one direction? Why is time different to space e.g. why do conservation laws operate over time & not space? 3 / 17

Leaves give evidence of the direction of the wind – they don’t cause the wind. LIKEWISE Entropy gives evidence of direction of time – it doesn’t cause time evolution. 4 / 17

Leaves give evidence of the direction of the wind – they don’t cause the wind. LIKEWISE Entropy gives evidence of direction of time – it doesn’t cause time evolution. 5 / 17

Why is time different to space? Recall parity and time reversal Here ℏ=1 . 𝛿𝑥 Translations in space: ( 𝑃 = momentum) 𝑥 e −𝑖 𝑃 𝛿𝑥 |𝑥 = |𝑥+𝛿𝑥 ℙ −𝟏 Parity inversion ℙ inverts 𝑥, 𝑦, & 𝑧 axes 𝛿𝑥 −𝛿𝑥 ℙ −𝟏 |𝑥 = |−𝑥 ℙ 𝑥 ℙ e −𝑖 𝑃 𝛿𝑥 ℙ −1 |𝑥 = |𝑥−𝛿𝑥 [Wigner, Group theory (1959)] 𝕋 −𝟏 𝛿𝑡 𝑡 −𝛿𝑡 𝕋 Translations in time: ( 𝐻 = Hamiltonian) e −𝑖 𝐻 𝛿𝑡 |𝜓(𝑡) = |𝜓(𝑡+𝛿𝑡) Time reversal 𝕋 inverts the 𝑡 axis: 𝕋 −𝟏 |𝜓(𝑡) = |𝜓(−𝑡) 𝕋 e −𝑖 𝐻 𝛿𝑡 𝕋 −1 |𝜓(𝑡) = |𝜓(𝑡−𝛿𝑡) More changes in direction  more opportunity to see symmetry violations Thus we want zigzag paths like Wiener process or Brownian motion 6 / 17

T violation and the Schrödinger equation 𝜕 𝜕𝑡 𝜓 𝑡 =−𝑖 𝐻 𝜓 𝑡 𝜓 𝑡 0 𝜓 𝑡 𝑡 e −𝑖 𝐻 (𝑡− 𝑡 0 ) solution 𝜓 𝑡 = e −𝑖 𝐻 (𝑡− 𝑡 0 ) 𝜓 𝑡 0 Time reversal: 𝜕 𝜕𝑡 𝕋 𝜓 𝑡 =+𝑖 𝕋 𝐻 𝕋 −1 𝕋 𝜓 𝑡 Write as 𝜙 𝑡 𝜙 𝑡 1 𝑡 e +𝑖 𝕋 𝐻 𝕋 −1 (𝑡− 𝑡 1 ) 𝜕 𝜕𝑡 𝜙 −𝑡 =+𝑖 𝕋 𝐻 𝕋 −1 𝜙(−𝑡) solution 𝜙 −𝑡 = e +𝑖 𝕋 𝐻 𝕋 −1 (𝑡− 𝑡 1 ) 𝜙 𝑡 1 Special case of T symmetry 𝕋 𝐻 𝕋 −1 = 𝐻 𝜓 𝑡 0 𝜓 𝑡 𝑡 e −𝑖 𝐻 (𝑡− 𝑡 0 ) 𝜙 𝑡 𝜙 𝑡 1 𝑡 e +𝑖 𝐻 (𝑡− 𝑡 1 ) 7 / 17

𝕋 e −𝑖 𝐻 (𝑡− 𝑡 0 ) e +𝑖 𝕋 𝐻 𝕋 −1 (𝑡− 𝑡 1 ) 𝑡 𝑡 Experiment: BarBar Collaboration, SLAC, Stanford University J. P. Lees et al., Phys. Rev. Lett. 109, 211801 (2012). Detailed description: Bernabeu et al, JETP Lett. 2012, 64 (2012). 𝕋 𝑡 e −𝑖 𝐻 (𝑡− 𝑡 0 ) 𝑡 e +𝑖 𝕋 𝐻 𝕋 −1 (𝑡− 𝑡 1 ) 8 / 17

Why is time different to space? Relativity: time and space have equal footing - they are essentially interconvertible 𝑡 𝑡 worldlines of particles spacetime background 𝑑𝑠 2 = 𝑑𝑡 2 − 𝑑𝑥 2 − 𝑑𝑦 2 − 𝑑𝑧 2 = 𝑑 𝑡 2 − 𝑑 𝑥 2 − 𝑑 𝑦 2 − 𝑑 𝑧 2 QFT 𝑥 However 𝑥 Dynamics (relativistic, QFT, quantum, classical): - mass / energy can be localised in space but not time. Conservation laws (energy, momentum, lepton number,...) apply over time but not over space. e.g. Quantum Mechanics: 𝜓(𝑥) 𝑥 wave function over space 𝜙(𝑡) 𝑡 mass not conserved wave function over time no equation of motion But symmetry between space and time implies this possibility 9 / 17

What if... a quantum theory gave space and time an equal footing at a fundamental level and differences between them arose phenomenologically? fundamental level 𝑥 𝜓(𝑥,𝑡) 𝑡 mass not conserved equal footing observed ?? 𝑥 𝜓(𝑥,𝑡) 𝑡 mass conserved ordered sequence equation of motion phenomenological details state is localised in time and space ? differences arise K0  e+  + e Need to look at phenomenology of generators of translations in space and time: kaon decay the Hamiltonian generates translations in time the momentum operator generates translations in space C,P,T discrete symmetries are violated by the Hamiltonian only! We need to include the P and T operations explicitly in our theory. 10 / 17

Gaussian distribution Prescription: 1 Express the state as superposition of zigzag paths fundamental level 𝑥 𝜓(𝑥,𝑡) 𝑡 mass not conserved equal footing 1-D Wiener process 𝜓 𝑥 Gaussian 𝑡 for reversals use localised  finite variance Gaussian distribution ℙ 𝕋 2 Include fundamental limits in precision Imagine a theory predicts a limit of a sequence 𝑎 1 , 𝑎 2 , 𝑎 3 , …, 𝑎 𝑛 , … where 𝑎 𝑛 → 𝑎 ∞ 𝑎 𝑛 :𝑛>𝑁 + n 𝑎 𝑛 𝑎 ∞ Δ𝑎 N resolution limit tail Let Δ𝑎 be the resolution limit. Then the tail of the sequence is a better representation than the limit point 𝑎 ∞ . 3 Consider violations of discrete symmetries 11 / 17

Quantum states in space 1 Express state as superposition of zigzag paths Consider a “galaxy” in 1D space Centre of mass position: Total momentum: Commutator: Consider Gaussian pure state for position: (non-relativistic quantum mechanics) 𝑋 , 𝑋 𝑥 X =𝑥 𝑥 X step size 𝛿𝑥= 𝜎 𝑛 random paths |𝜓⟩ 𝜎 𝑥 𝑃 𝑋 , 𝑃 =𝑖 use ℙ for reversals 𝜓 ∝∫𝑑𝑥 e − 𝑥 2 4 𝜎 2 𝑥 X ∝ lim 𝑛→∞ ℙ e −𝑖 𝑃 𝛿𝑥 ℙ −𝟏 + e −𝑖 𝑃 𝛿𝑥 2 0 X 𝑛 expanding … 𝑛 gives superposition of random paths 12 / 17

2 Account for fundamental limits in precision step size 𝛿𝑥= 𝜎 𝑛 |𝜓⟩ 𝜎 𝑥 random paths As 𝑛→∞ , the step size 𝛿𝑥 will eventually become smaller than any fundamental resolution limit Δ𝑥 for distinguishing spatial separations, e.g. Δ𝑥≈1.6× 10 −35 m = Planck length. use ℙ for reversals For 𝑛>𝑁 we have 𝛿𝑥<Δ𝑥 . Hence, define the set of physically indistinguishable states: 𝚿= |𝜓〉 𝑛 : 𝑛>𝑁 step size 𝛿𝑥<Δ𝑥 random paths 𝜓 𝑛 𝑥 = the tail of the sequence where 𝜓 𝑛 ∝ ℙ e −𝑖 𝑃 𝛿𝑥 ℙ −𝟏 + e −𝑖 𝑃 𝛿𝑥 2 0 X 𝑛 3 Consider parity violation No change. The galaxy is localised in space, has zigzag paths, & finite precision 13 / 17 13

Quantum states in time ? 1 & 2: zigzag paths and finite precision Use the same construction for a galaxy distributed over time. I.e. we have a set of states, all of equal status: Υ 𝑛 𝑡 ? 𝚼= |Υ〉 𝑛 : 𝑛>𝑁 use where Υ 𝑛 ∝ 𝕋 e −𝑖 𝐻 𝛿𝑡 𝕋 −𝟏 + e −𝑖 𝐻 𝛿𝑡 2 𝜙 T 𝑛 𝕋 random paths for reversals step size 𝛿𝑡<Δ𝑡 𝛿𝑡= 𝜎 𝑛 <Δ𝑡 step size e.g. Δ𝑡≈5.4× 10 −44 s Planck time 3 Consider T symmetry & T violation mass is not conserved random paths Υ 𝑛 𝑡 𝜎 T symmetry 𝕋 𝐻 𝕋 −𝟏 = 𝐻 no equation of motion (antilinear) 𝕋 𝑖 𝕋 −1 =−𝑖 𝕋 e −𝑖 𝐻 𝛿𝑡 𝕋 −1 = e 𝑖 𝐻 𝛿𝑡 (reversal) The galaxy is localised in time, has zigzag paths, & finite precision 14 / 17 14

Υ 𝑛 ∝ e −𝑖 𝕋 𝐻 𝕋 −𝟏 𝛿𝑡 + e −𝑖 𝐻 𝛿𝑡 2 𝜙 T 𝑛 ⋯ ⋯ ⋯ ⋯ T violation (T violation could be due to kaon, B meson decay, or perhaps even the Higgs field.) 𝕋 𝐻 𝕋 −𝟏 ≠ 𝐻 𝕋 𝐻 𝕋 −𝟏 , 𝐻 =𝑖𝜆 (minimalist) Get interference between different paths to the same point in time. 𝑡 (peak) = 2𝜋 𝑛 𝜎𝜆 𝚼= |Υ〉 𝑛 :𝑛> 𝑁 Υ 𝑛 (ordered set) 𝕋 𝐻 𝕋 −𝟏 𝐻 where Υ 𝑛 ∝ e −𝑖 𝕋 𝐻 𝕋 −𝟏 𝛿𝑡 + e −𝑖 𝐻 𝛿𝑡 2 𝜙 T 𝑛 ⋯ ⋯ ⋯ ⋯ 𝑡 −𝑡 (peak) 𝑡 (peak) 𝛿𝑡= 𝜎 𝑛 >Δ𝑡 t Υ 𝑛 arXiv: 1502.04012, (2015) destructive interference For any given time 𝑡 there is a state Υ 𝑛 ∈𝚼 with 𝑛= 𝑡 2 𝜎𝜆 2𝜋 2 which represents the galaxy and its mass existing at that time (and −𝑡).  mass is conserved 15 / 17 15

Emergence of conventional Quant. Mech. Coarse graining over time 𝑡 −𝑡 ⋯ Υ + (𝑡) Υ − (𝑡) 𝚼= |Υ〉 𝑛 :𝑛> 𝑁 Imagine time resolution is much larger than Δ𝑡. Then |Υ〉 𝑛 becomes a continuous function of 𝑡 (peak) ≈𝑡 : 𝚼= |Υ(𝑡)〉 : 𝑡>𝑇 𝑇= 2𝜋 𝑁 𝜎𝜆 Υ(t) ∝ Υ − 𝑡 + Υ + 𝑡 , Separate T invariant (i) and T violating (v) parts: 𝐻 = 𝐻 (i) ⊗ 𝟏 v + 𝟏 i ⊗ 𝐻 (v) Schrodinger equation 𝑑 𝑑𝑡 Υ + 𝑡 ≈−𝑖 𝐻 i ⊗ 𝟏 v + 𝟏 i ⊗ 𝑯 𝐩𝐡𝐞𝐧 𝐯 | Υ + 𝑡 〉 𝑯 𝐩𝐡𝐞𝐧 (𝐯) = 𝐻 (v) 𝑎 + − 𝕋 𝐻 v 𝕋 −1 𝑎 − 𝑎 ± = 𝜃±2𝜋 4𝜋 where 𝜃=2.23𝜋 Experimental evidence: 𝑯 𝐩𝐡𝐞𝐧 (𝐯) ≠ 𝐻 (v) 16 / 17

Found. Phys. 41, 1569 (2011) Summary Summary Found. Phys. 45, 691 (2015) Proc. R. Soc. A 472, (2016) New quantum formalism – states in time & space have same footing: explicitly includes parity ℙ and time 𝕋 inversion operations – exposes violations states are represented as superpositions of paths (c.f. Feynman path integral) conservation of mass is not assumed 𝑡 −𝑡 ⋯ 𝕋 𝐻 𝕋 −𝟏 𝐻 T symmetry T violation 𝑡 differences between time and space emerge phenomenologically ordered set of states – gives a direction to time (double headed arrow) translates mass over time – conservation of mass (if 𝐻 , 𝕋 𝐻 𝕋 −𝟏 allow it) interpret T violation as the origin of dynamics. 17 / 17