The Physical Basis of The Direction of Time (The Frontiers ...

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22 2 The Time Arrow of Radiation t 2 t t 2 t P P t 1 t 1 Fig. 2.2. Two representations of the same electromagnetic potential at time t by means of retarded or advanced Green’s functions. They require data on partial boundaries (indicated by solid lines) corresponding to an initial or a final value problem, respectively spacetime boundary required for determining the potential at time t assumes specific forms indicated in Fig. 2.2. Hence, the same potential can be written according to one or the other RHS of A µ = source term + boundary terms = A µ ret + A µ in = A µ adv + Aµ out . (2.8) For example, A µ in is here that solution of the homogeneous equations which coincides with A µ for t = t 1 . A µ ret and A µ adv vanish by definition for t = t 1 or t = t 2 , respectively. Any field can therefore be described equivalently by an initial or a final value problem – with arbitrary boundary conditions. This result reflects the T -symmetry of the laws, while phenomenological causality is often used as an ad hoc argument for choosing G ret rather than G adv . However, two free boundary conditions in the mixed form of Fig. 2.1 would in general not be consistent with one another, even if individually incomplete (see also Sects. 2.4 and 5.3). Retarded and advanced fields formally resulting from past and future sources, respectively, do not add independently (as sometimes assumed to describe a conjectured retro-causation) – they just contribute to different (or mixed) representations of the same field. In field theory, no (part of the) field ‘belongs to’ a certain source (in contrast to specific action-at-a-distance theories). Sources determine only the difference A µ out − A µ in – similar to T/i = S − 1 in the interaction picture of the S-matrix. As can be seen from (2.8), this difference is identical to A µ ret − A µ adv . In causal language, whereA µ in is regarded as given, the source ‘creates’ precisely its retarded field that has to be added to A µ in in the future of the source (where A µ adv = 0). Physically, spatial boundary conditions represent an interaction with the (often uncontrollable) spatial environment. For infinite spatial volume (V = R 3 ), when the light cone cannot reach ∂V within finite time t − t 1 ,orina closed universe, one loses this boundary term in (2.7), and thus obtains the pure initial value problem (for t>t 1 ),
2.1 Retarded and Advanced Form of the Boundary Value Problem 23 ∫ t ∫ A µ = A µ ret + A µ in ≡ G ret (r,t; r ′ ,t ′ )j µ (r ′ ,t ′ )d 3 r ′ dt ′ (2.9) t 1 R 3 + 1 [ ] G ret (r,t; r 4π ∫R ′ ,t 1 )∂ t1 A µ (r ′ ,t 1 ) − A µ (r ′ ,t 1 )∂ t1 G ret (r,t; r ′ ,t 1 ) d 3 r ′ , 3 and correspondingly the pure final value problem (t 0, this is the limit a → 0forfixedak and m/a. The crucial restriction to ‘attractive’ forces (k > 0) may here appear surprising, since Newton’s equations are always deterministic, and allow one to pose initial conditions regardless of the type
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Page 4 and 5: H. Dieter Zeh THE PHYSICAL BASIS OF
Page 6 and 7: Preface Four previous editions of t
Page 8 and 9: Contents Introduction .............
Page 10 and 11: Introduction The asymmetry of Natur
Page 12 and 13: Introduction 3 tion of time (thus a
Page 14 and 15: Introduction 5 1. Radiation. In mos
Page 16 and 17: Introduction 7 language is loaded w
Page 18 and 19: Introduction 9 between temporal and
Page 20 and 21: 12 1 The Physical Concept of Time i
Page 22 and 23: 14 1 The Physical Concept of Time r
Page 24 and 25: 16 1 The Physical Concept of Time a
Page 26 and 27: 18 2 The Time Arrow of Radiation Th
Page 28 and 29: 20 2 The Time Arrow of Radiation by
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Page 34 and 35: 26 2 The Time Arrow of Radiation wa
Page 36 and 37: 28 2 The Time Arrow of Radiation th
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Page 46 and 47: 38 2 The Time Arrow of Radiation 'i
Page 48 and 49: 40 3 The Thermodynamical Arrow of T
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72 3 The Thermodynamical Arrow of T
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100 4 The Quantum Mechanical Arrow
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136 5 The Time Arrow of Spacetime G
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Epilog Four weeks before his death,
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Epilog 201 The role of time is very
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Appendix: A Simple Numerical Toy Mo
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Notebook: A Toy Model Appendix: A S
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4. Two-Time Boundary Condition Appe
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References References marked with (
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References 211 Bläsi, B., and Hard
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References 213 Davies, P.C.W., and
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References 215 Gerlach, U.H. (1988)
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References 217 Horwich, P. (1987):
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References 219 Levine, H., Moniz, E
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References 221 Pearle, Ph. (1976):
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References 223 Smoluchowski, M. Rit
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References 225 Zeh, H.D. (1971): On
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Index absorber, 18, 24-28, 36-38, 5
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Index 229 forward determinism, 66,
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Index 231 stochastic process, see d
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