Griffiths R.B. Consistent Quantum Theory

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374 Bibliography

described in Sec. 7.5.7 of Born and Wolf (1980). For a discussion of neutron

interference effects, see the review article by Greenberger (1983).

Ch. 14. Dependent (contextual) events

Branch-dependent histories, Sec. 14.2, were introduced by Gell-Mann and Hartle

(1993).

Ch. 16. Quantum reasoning

The importance of having appropriate rules for quantum reasoning has been under-

stood (at least in some quarters) ever since the work of Birkhoff and von Neumann

(1936), and has been stressed by Omn

es (1992, 1994, 1999). The formulation

given here is based on Grifﬁths (1996, 1998a).

Chs. 17 and 18. Measurements

A very readable account of the problems encountered in trying to base the inter-

pretation of quantum theory upon measurements will be found in Wigner (1963).

Despite a great deal of effort in the quantum foundations community there has been

essentially no progress in solving these problems; see the protest by Bell (1990)

and the pessimistic assessment by Mittelstaedt (1998).

Chapter VI of von Neumann (1932) contains his theory of quantum measure-

ments. That the absence of some occurrence can constitute a “measurement” was

pointed out by Renninger (1960), although according to Jammer (1974), p. 495,

this difﬁculty was known much earlier. The L

uders rule is found in L

uders (1951).

The nondestructive measurement apparatus in Fig. 18.2 was inspired by the one

in Fig. 5-3 of Feynman et al. (1965).

Ch. 19. Coins and counterfactuals

The importance of counterfactuals for understanding quantum theory has been

stressed by d’Espagnat (1984, 1989). Philosophers have analyzed counterfactual

reasoning with somewhat inconclusive results; the work of Lewis (1973, 1986) is

often cited. The approach used in this chapter extends Grifﬁths (1999a).

Ch. 20. Delayed choice paradox

The paradox in the form considered here is due to Wheeler (1978, 1983). The

analysis uses material from Grifﬁths (1998b). For experimental realizations see

Alley et al. (1983, 1987) and Hellmuth et al. (1987).

Bibliography 375

Ch. 21. Indirect measurement paradox

Aside from dramatization, the paradox stated here is that of Elitzur and Vaidman

(1993). However, the basic idea seems to go back much earlier; see p. 495 of

Jammer (1974). A somewhat similar paradox was put forward by Hardy (1992b).

The term “interaction-free” appears to have originated with Dicke (1981).

Ch. 22. Incompatibility paradoxes

The impossibility of simultaneously assigning values to all quantum variables was

pointed out by Bell (1966) and by Kochen and Specker (1967). See the helpful

discussion of these and other results in Mermin (1993). The two-spin paradox of

Mermin (1990) was inspired by earlier work by Peres (1990). See Mermin (1993)

for a very clear presentation of this and related paradoxes.

The discussion of truth functionals in this chapter draws on Grifﬁths (2000a,b).

The paradox of three boxes in Sec. 22.5 was introduced by Aharonov and Vaidman

(1991); the discussion given in this chapter extends that in Grifﬁths (1996).

Ch. 23. Singlet state correlations

The original paper by Einstein, Podolsky, and Rosen (1935) has given rise to an

enormous literature. A few of the items that I myself have found helpful are: Fine

(1986), Greenberger et al. (1990), and Hajek and Bub (1992).

Bohm’s study of the EPR problem using a singlet state of two spin-half particles

appeared in Ch. 22 of Bohm (1951). The analysis of spin correlations given here

extends an earlier discussion in Grifﬁths (1987).

Ch. 24. EPR paradox and Bell inequalities

See the references above for Ch. 23.

Bohm’s hidden variable theory was introduced in Bohm (1952). For recent for-

mulations, see Bohm and Hiley (1993) and Berndl et al. (1995). A serious prob-

lem with the Bohm theory was pointed out by Englert et al. (1992); see Grifﬁths

(1999b) for references to the ensuing discussion.

Mermin’s model for hidden variables appeared in Mermin (1981, 1985). A num-

ber of Bell’s papers have been reprinted in Bell (1987), which is an excellent source

for his work. These include the original inequality paper, Bell (1964). The CHSH

inequality was published by Clauser et al. (1969), and a derivation will be found in

Ch. 4 of Bell (1987). There is a vast literature devoted to Bell inequalities and their

signiﬁcance. In addition to the sources already mentioned, the reader may ﬁnd it

376 Bibliography

helpful to consult Ch. 4 of Redhead (1987), Shimony (1990), and Greenberger et

al. (1990).

Ch. 25. Hardy’s paradox

Hardy (1992a) is the original publication of his paradox. Mermin (1994) gives a

very clear exposition of the basic idea. The GHZ paradox is explained in Green-

berger et al. (1990). The counterfactual analysis in Sec. 25.5 extends Grifﬁths

(1999a).

Ch. 26. Decoherence and the classical limit

A great deal has been written on the topic of decoherence and its relationship to

the emergence of classical physics from quantum theory. For an introduction to

the subject, see Zurek (1991). The book by Giulini et al. (1996) has contributions

from diverse points of view and extensive references to earlier work. For work

from a perspective close to the point of view found here, see Gell-Mann and Hartle

(1993), Omn

es (1999) (which gives references to earlier work), and Brun (1993,

1994).

Forthe argument of Omn

es on the relationship of classical and quantum mechanics

referred to in Sec. 26.6, see Chs. 10 and 11 of Omn

es (1999), and his references to

earlier work.

Ch. 27. Quantum theory and reality

On the difﬁculty of observing MQS states, Sec. 27.4, see Omn

es (1994), Ch. 7,

Sec. 8.

References

Note: WZ refers to the source book by Wheeler and Zurek (1983).

Aharonov, Y. and L. Vaidman (1991). Complete description of a quantum system at a given

time. J. Phys. A 24, 2315.

Alley, C. O., O. Jakubowicz, C. A. Steggerda, and W. C. Wickes (1983). A delayed random

choice quantum mechanics experiment with light quanta, in Proc. Int. Symp. Founda-

tions of Quantum Mechanics, ed. S. Kamefuchi et al. (Physical Society of Japan,

Tokyo), p. 158.

Alley, C. O., O. Jakubowicz, and W. C. Wickes (1987). Results of the delayed-random-

choice quantum mechanics experiment with light quanta, in Proc. 2nd Int. Symp.

Foundations of Quantum Mechanics, ed. M. Namiki (Physical Society of Japan,

Tokyo), p. 36.

Bell, J. S. (1964). On the Einstein Podolsky Rosen paradox. Physics 1, 195. Reprinted in

Ch. 2 of Bell (1987).

Bell, J. S. (1966). On the problem of hidden variables in quantum mechanics. Rev. Mod.

Phys. 38, 447.

Bell, J. S. (1987). Speakable and Unspeakable in Quantum Mechanics (Cambridge Uni-

versity Press, Cambridge, UK).

Bell, J. S. (1990). Against measurement, in Sixty-Two Years of Uncertainty, edited by

A. I. Miller (Plenum Press, New York), p. 17.

Berndl, K., M. Daumer, D. D

urr, S. Goldstein, and N. Zangh

ı (1995). A survey of Bohmian

mechanics. Nuovo Cimento B 110, 737.

Birkhoff, G. and J. von Neumann (1936). The logic of quantum mechanics. Annals Math.

37, 823. Reprinted in John von Neumann Collected Works, ed. A. H. Taub (Macmillan,

New York, 1962), Vol. IV, p. 105.

Blank, B., P. Exner, and M. Havl

cek (1994). Hilbert Space Operators in Quantum Physics

(American Institute of Physics, New York).

Bohm, D. (1951). Quantum Theory (Prentice Hall, New York).

Bohm, D. (1952). A suggested interpretation of the quantum theory in terms of “hidden”

variables, I and II. Phys. Rev. 85, 166. Reprinted in WZ p. 369.

Bohm, D. and B. J. Hiley (1993). The Undivided Universe (Routledge, London).

Bohr, N. (1951). Discussion with Einstein on epistemological problems in atomic physics,

in Albert Einstein: Philosopher Scientist, 2d edition, ed. P. A. Schilpp (Tudor Pub-

lishing, New York), p. 199. Reprinted in WZ p. 9.

377

378 References

Born, M. (1926). Zur Quantenmechanik der Stoßvorg

ange. Z. Phys. 37, 863. An English

translation is in WZ, p. 52.

Born, M. and E. Wolf (1980). Principles of Optics, 6th edition (Pergamon Press, Oxford).

Bransden, B. H. and C. J. Joachain (1989).Introduction to Quantum Mechanics (Longman,

London).

Brun, T. A. (1993). Quasiclassical equations of motion for nonlinear Brownian systems.

Phys. Rev. D 47, 3383.

Brun, T. A. (1994). Applications of the Decoherence Formalism, PhD dissertation, Cali-

fornia Institute of Technology.

Clauser, J. F., M. A. Horne, A. Shimony, and R. A. Holt (1969). Proposed experiment to

test local hidden-variable theories. Phys. Rev. Lett. 23, 880.

Cohen-Tannoudji, C., B. Diu, and F. Lalo

e (1977). Quantum Mechanics (Hermann, Paris).

DeGroot, M. H. (1986). Probability and Statistics, 2d edition (Addison-Wesley, Reading,

Massachusetts).

d’Espagnat, B. (1984). Nonseparability and the tentative descriptions of reality. Phys.

Repts. 110, 201.

d’Espagnat, B. (1989). Reality and the Physicist (Cambridge University Press, Cambridge,

UK).

Dicke, R. H. (1981). Interaction-free quantum measurements: a paradox? Am. J. Phys. 49,

925.

Dirac, P. A. M. (1958). The Principles of Quantum Mechanics, 4th edition (Oxford Uni-

versity Press, Oxford).

Dowker, F. and A. Kent (1996). On the consistent histories approach to quantum mechan-

ics. J. Stat. Phys. 82, 1575.

Einstein, A., B. Podolsky, and N. Rosen (1935). Can quantum-mechanical description of

physical reality be considered complete? Phys. Rev. 47, 777.

Elitzur, A. C. and L. Vaidman (1993). Quantum mechanical interaction-free measurements.

Found. Phys. 23, 987.

Englert, B. G., M. O. Scully, G. Sussmann, and H. Walther (1992). Surrealistic Bohm

trajectories. Z. Naturforsch. 47a, 1175.

Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd

edition (John Wiley & Sons, New York).

Fetter, A. L. and J. D. Walecka (1971). Quantum Theory of Many-Particle Systems

(McGraw-Hill, New York).

Feynman, R. P., R. B. Leighton, and M. Sands (1965). The Feynman Lectures on Physics,

Vol. III (Addison-Wesley, Reading, Massachusetts).

Fine, A. (1986). The Shaky Game: Einstein Realism and the Quantum Theory (University

of Chicago Press, Chicago). A similar perspective will be found in the 2nd edition

(1996).

Gell-Mann, M. and J. B. Hartle (1990). Quantum mechanics in the light of quantum cos-

mology, in Complexity, Entropy, and the Physics of Information, ed. W. Zurek (Addi-

son Wesley, Reading, Massachusetts), p. 425; also in Proceedings of the 25th Inter-

national Conference on High Energy Physics, Singapore, 1990, eds. K. K. Phua and

Y. Yamaguchi (World Scientiﬁc, Singapore).

Gell-Mann, M. and J. B. Hartle (1993). Classical equations for quantum systems. Phys.

Rev. D 47, 3345.

Gieres, F. (2000). Mathematical surprises and Dirac’s formalism in quantum mechanics.

Rep. Prog. Phys. 63, 1893.

Giulini, D., E. Joos, C. Kiefer, J. Kupsch, I.-O. Stamatescu, and H. D. Zeh (1996). De-

coherence and the Appearance of a Classical World in Quantum Theory (Springer,

References 379

Berlin).

Greenberger, D. M. (1983). The neutron interferometer as a device for illustrating the

strange behavior of quantum systems. Rev. Mod. Phys. 55, 875.

Greenberger, D. M., M. A. Horne, A. Shimony, and A. Zeilinger (1990). Bell’s theorem

without inequalities. Amer. J. Phys. 58, 1131.

Grifﬁths, R. B. (1984). Consistent histories and the interpretation of quantum mechanics.

J. Stat. Phys. 36, 219.

Grifﬁths, R. B. (1987). Correlations in separated quantum systems: a consistent history

analysis of the EPR problem. Am. J. Phys. 55, 11.

Grifﬁths, R. B. (1996). Consistent histories and quantum reasoning. Phys. Rev. A 54, 2759.

Grifﬁths, R. B. (1998a). Choice of consistent family, and quantum incompatibility. Phys.

Rev. A 57, 1604.

Grifﬁths, R. B. (1998b). Consistent histories and quantum delayed choice. Fortschr. Phys.

46, 741.

Grifﬁths, R. B. (1999a). Consistent quantum counterfactuals. Phys. Rev. A 60,5.

Grifﬁths, R. B. (1999b). Bohmian mechanics and consistent histories. Phys. Lett. A 261,

227.

Grifﬁths, R. B. (2000a). Consistent histories, quantum truth functionals, and hidden vari-

ables. Phys. Lett. A 265, 12.

Grifﬁths, R. B. (2000b). Consistent quantum realism: A reply to Bassi and Ghirardi. J.

Stat. Phys. 99, 1409.

Gudder, S. (1989). Realism in quantum mechanics. Found. Phys. 19, 949.

Hajek, A. and J. Bub (1992). EPR. Found. Phys. 22, 313.

Halmos, P. R. (1957). Introductionto Hilbert Space and the Theory of Spectral Multiplicity,

2nd edition (Chelsea Publishing, New York).

Halmos, P. R. (1958). Finite-Dimensional Vector Spaces, 2nd edition (Van Nostrand,

Princeton).

Hardy, L. (1992a). Quantum mechanics, local realistic theories and Lorentz-invariant real-

istic theories. Phys. Rev. Lett. 68, 2981.

Hardy, L. (1992b). On the existence of empty waves in quantum theory. Phys. Lett. A 167,

11.

Hellmuth, T., H. Walther, A. Zajonc, and W. Schleich (1987). Delayed-choice experiments

in quantum interference. Phys. Rev. A 35, 2532.

Horn, R. A. and C.R. Johnson (1985). Matrix Analysis (Cambridge University Press, Cam-

bridge, UK).

Horn, R. A. and C. R. Johnson (1991). Topics in Matrix Analysis (Cambridge University

Press, Cambridge, UK).

Isham, C. J. (1994). Quantum logic and the histories approach to quantum theory. J. Math.

Phys. 35, 2157.

Jammer, M. (1974). The Philosophy of Quantum Mechanics (Wiley, New York).

Kochen, S. and E. P. Specker (1967). The problem of hidden variables in quantum mechan-

ics. J. Math. Mech. 17, 59.

Lewis, D. (1973). Counterfactuals (Harvard University Press, Cambridge, Massachusetts).

Lewis, D. (1986). Philosophical Papers, Vol. 2 (Oxford University Press, New York).

uders, G. (1951).

Uber die Zustands

anderung durch den Meßprozeß. Ann. Phys. (Leipzig)

8, 322.

utkepohl, H. (1996). Handbook of Matrices (Wiley, New York).

McElwaine, J. N. (1996). Approximate and exact consistency of histories. Phys. Rev. A 53,

2021.

380 References

Mermin, N. D. (1981). Bringing home the atomic world: quantum mysteries for anybody.

Am. J. Phys. 49, 940.

Mermin, N. D. (1985). Is the moon there when nobody looks? Reality and the quantum

theory. Physics Today 38, April, p. 38.

Mermin, N. D. (1990). Simple uniﬁed form for the major no-hidden-variables theorems.

Phys. Rev. Lett. 65, 3373.

Mermin, N. D. (1993). Hidden variables and the two theorems of John Bell. Rev. Mod.

Phys. 65, 803.

Mermin, N. D. (1994). Quantum mysteries reﬁned. Am. J. Phys. 62, 880.

Meyer, D. A. (1996). On the absence of homogeneous scalar unitary cellular automata.

Phys. Lett. A 223, 337.

Merzbacher, E. (1998). Quantum Mechanics, 3nd edition (Wiley, New York).

Mittelstaedt, P. (1998). The Interpretation of Quantum Mechanics and the Measurement

Process (Cambridge University Press, Cambridge, UK).

Nielsen, M. A. and I. L. Chuang (2000). Quantum Computation and Quantum Information

(Cambridge University Press, Cambridge, UK).

Omn

es, R. (1992). Consistent interpretations of quantum mechanics. Rev. Mod. Phys. 64,

339.

Omn

es, R. (1994). The Interpretation of Quantum Mechanics (Princeton University Press,

Princeton).

Omn

es, R. (1999). Understanding Quantum Mechanics (Princeton University Press,

Princeton).

Peres, A. (1990). Incompatible results of quantum measurements. Phys. Lett. A 151, 107.

Redhead, M. (1987). Incompleteness, Nonlocality, and Realism (Oxford University Press,

Oxford).

Reed, M. and B. Simon (1972). Methods of Modern Mathematical Physics. I: Functional

Analysis (Academic Press, New York).

Renninger, M. (1960). Messungen ohne St

orung des Meßobjekts. Z. Phys. 158, 417.

Ross, S. M. (2000). Introduction to Probability Models, 7th edition (Harcourt Academic

Press, San Diego).

Schr

odinger, E. (1935). Die gegenw

artige Situation in der Quantenmechanik. Naturwis-

senschaften 23 807, 823, 844. An English translation is in WZ, p. 152.

Schwabl, F. (1992). Quantum Mechanics (Springer-Verlag, Berlin).

Shankar, R. (1994). Principles of Quantum Mechanics, 2nd edition (Plenum Press, New

York).

Shimony, A. (1990). An exposition of Bell’s theorem, in Sixty-Two Years of Uncertainty,

ed. A. I. Miller, (Plenum Press, New York) p. 33. Reprinted in A. Shimony, Search

for a Naturalistic World View, Vol. II (Cambridge University Press, Cambridge, UK,

1993), p. 90.

Strang, G. (1988). Linear Algebra and Its Applications, 3rd edition (Harcourt, Brace and

Jovanovich, San Diego).

von Neumann, J. (1932). Mathematische Grundlagen der Quantenmechanik (Springer-

Verlag, Berlin). English translation: Mathematical Foundations of Quantum Mechan-

ics (Princeton University Press, Princeton, 1955).

Wannier, G. H. (1966). Statistical Physics (Wiley, New York).

Wheeler, J. A. (1978). The “past” and the “delayed-choice” double-slit experiment, in

Mathematical Foundations of Quantum Theory, ed. A. R. Marlow (Academic Press,

New York), p. 9.

Wheeler, J. A. (1983). Law without law, in WZ, p. 182.

Wheeler, J. A. and W. H. Zurek (editors) (1983). Quantum Theory and Measurement

References 381

(Princeton University Press).

Wigner, E. P. (1963). The problem of measurement. Am. J. Phys. 31,6.

Zurek, W. H. (1991). Decoherence and the transition from quantum to classical. Phys.

Today 44, Oct., p. 36.

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Index

I =



|jj|,38

Pr():Pr(E),73

Pr(

):Pr(A

B),74

Tr():Tr(A),42

():Tr

(D),90

|: |ψ ,27

|: ω|,29

|: ω|ψ,29

||: φ|A|ψ,31

: V,79–80

, 

: A,

A

, 210

, 

: A,

A

, 214

[]:[φ] =|φφ|/φ|φ,34

∼: ∼P =

P = I − P,35

∧: P ∧ Q,58

∨: P ∨ Q,58

(: H

( H

, F

( F

, 112

⊗: A ⊗ B, |a⊗|b, A ⊗ B,82–87

†, see dagger operation

adjoint, 33–34,39

Aharonov, Y., 375

alpha decay (toy model), 105–107, 131–134, 171–173

Born rule, 132

consistent family, 172

with detector, 131, 172

exponential decay, 133

time of decay, 172

unitary dynamics, 106–107, 132

AND, see conjunction

antilinear, 28,30

approximate, see consistency conditions, energy,

momentum, position

aspects, 365

atoms, see interferometer for atoms

average, 79–80

from density matrix, 80, 204

of indicator, 79

as linear functional, 79

of observable, 130

quantum, 80

badminton, 265

base, see contextual

baseball, 108

basis, see orthonormal

beam splitter, 159

classical analogy, 245, 248

and detectors, 243–246

conditional probabilities, 245

consistent family, 245

unitary dynamics, 243

inside box, 252–255

movable, 263, 273–282, 343–348