1. Quantum asymmetry between time and space &
the origin of dynamics
Joan Vaccaro Centre for Quantum Dynamics
Griffith University
Brisbane
Australia
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2. DOI: /rspa
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3. 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?
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4. 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.
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5. 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.
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6. 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
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7. 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 )
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8. 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, (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 )
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9. 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
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10. 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.
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11. 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
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12. 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 𝑥 X
∝ lim 𝑛→∞ ℙ e −𝑖 𝑃 𝛿𝑥 ℙ −𝟏 + e −𝑖 𝑃 𝛿𝑥 X 𝑛
expanding … 𝑛 gives superposition of random paths
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13. 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 X 𝑛
3 Consider parity violation
No change.
The galaxy is localised in space, has zigzag paths, & finite precision
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14. 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 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
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15. 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 𝜙 T 𝑛 ⋯ ⋯ ⋯ ⋯ 𝑡
−𝑡 (peak)
𝑡 (peak)
𝛿𝑡= 𝜎 𝑛 >Δ𝑡 t
Υ 𝑛
arXiv: , (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
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16. 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)
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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.
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