In 'The Algebra of Microscopic Measurement' (1959), Julian Schwinger derived the finite-dimensional kinematic formalism of quantum mechanics - measurement symbols, transformation functions, and the probability law - from a symbolic transcription of idealized selective measurements. The paper restricts its exposition to physical quantities assuming only finitely many values, and its closing paragraphs announce a detailed formulation of quantum dynamics whose description of the interactions constituting measurement reproduces that symbolic characterization (including amplification of microscopic effects to macroscopic observation) and a geometry associated with the states of systems, while a footnote defers the significance of the appearance of complex numbers, which Schwinger conjectured to be an aspect of the fundamental matter-antimatter duality. The geometry-of-states direction was discharged by his companion paper 'The Geometry of Quantum States' (1960), and the extension to continuous spectra was supplied by 'The Special Canonical Group' (1960), both consolidated in his books. The surviving open core is the dynamical consistency requirement - the quantum measurement problem in Schwinger's operational formulation - together with the complex-number question: decoherence theory reproduces only the random-phase (nonselective) part of the symbolism from unitary dynamics, the probability law remains postulated or interpretation-dependent, and the conjectured matter-antimatter interpretation of complex conjugation has never been developed or tested. The problem stated here is this surviving core.

Background

Schwinger's measurement algebra is an operational reconstruction of quantum mechanics that takes the act of measurement, rather than waves, operators, or an equation of motion, as the primitive concept. In the source paper one considers idealized systems in which every physical quantity AA assumes only a finite number of distinct values a′a', and symbolizes by M(a′)M(a') the selective measurement that accepts systems exhibiting the value a′a' and rejects all others. Addition of symbols forms less specific filters; multiplication, read right to left, means performing measurements in succession. A selective measurement that accepts systems in the state a′′a'' and produces them in the state a′a' is written M(a′,a′′)M(a',a''), and its algebra - noncommutative multiplication with M(a′,a′′)M(a′′′,a′′′′)M(a',a'')M(a''',a'''') equal to δ(a′′,a′′′)M(a′,a′′′′)\delta(a'',a''')M(a',a''''), where δ\delta is the Kronecker symbol picking out equal values, and an N2N^2-dimensional linear algebra for NN states - reproduces matrix mechanics. A complex number ⟨a′∣b′⟩\langle a'|b'\rangle, the transformation function, relates incompatible complete descriptions aa and bb; requiring the probability p(a′,b′)=∣⟨a′∣b′⟩∣2p(a',b') = |\langle a'|b'\rangle|^2 to be real and nonnegative forces (b′∣a′)=(a′∣b′)∗(b'|a') = (a'|b')^*, which introduces complex numbers and the adjoint operation. A nonselective measurement, in which the apparatus functions but selects nothing, is represented by randomly phased measurement symbols - the algebraic expression of the uncontrollable disturbance produced by measurement. The paper's scope and closing remarks comprise: (i) the restriction to finitely many values is explicit, stated as a simplification of the exposition, and continuous spectra is not addressed; (ii) 'a detailed formulation of quantum dynamics must satisfy the consistency requirement that its description of the interactions that constitute measurement reproduces the symbolic characterizations that have emerged at this elementary stage', with explicit reference to the amplification of microscopic effects to the level of macroscopic observation; (iii) 'further analysis of the measurement algebra leads to a geometry associated with the states of systems'; and (iv) a footnote reports, from Schwinger's 'Spin, Statistics, and the TCP Theorem' (1958), the conjecture that 'the appearance of complex numbers, or their real equivalents, may be an aspect of the fundamental matter-antimatter duality', which the footnote defers as a question that 'can hardly be discussed at this stage'. In that 1958 paper (the TCP theorem asserts, for Lorentz-covariant quantum field theory with the spin-statistics connection, invariance under combined space-time reflection and charge conjugation, i.e., the correspondence of particles with antiparticles), Schwinger proposed that complex conjugation be given the physical interpretation of the change of description accompanying the substitution of an antimatter apparatus for a matter apparatus, and stated that quantum field theory could be reconstructed on this basis - a discussion he deferred to another occasion. The restriction (i) was lifted and direction (iii) carried out by the companion papers 'The Special Canonical Group' and 'The Geometry of Quantum States' (both 1960): the first obtains the continuous position-momentum degree of freedom as the limit of finite measurement algebras, supplying the continuous-spectra extension (with later rigorous caveats about collapse for continuous-spectrum observables analyzed by Srinivas, 1980, within standard quantum mechanics); the second derives the unitary geometry of states (including the Dirac bracket notation) and the metrical geometry of operator space from the measurement algebra extended by a null state. The unresolved target is the completion of the physical foundations of this operational program, and it is the operational form of the quantum measurement problem: deriving the outcome probabilities and state changes of measurement from the dynamics of system, apparatus, and environment rather than postulating them. Decoherence theory - the dynamical suppression of interference between macroscopically distinct apparatus states through environmental entanglement, reviewed by Zurek (2003) - reproduces the random-phase, nonselective part of the symbolism; the probability law itself remains postulated, with objective-collapse models (stochastic modifications of the Schrödinger equation, reviewed by Bassi et al., 2013) and new non-unitary evolution laws (the ETH-approach of Fröhlich and Pizzo, 2021, which claims to eliminate the measurement problem) as active but contested attempts rather than consensus closure.

Problem Statement

Complete the physical foundations of Schwinger's measurement-algebra program in the two items left open by the source paper and its 1960 companion papers ('The Geometry of Quantum States' having derived the unitary geometry of states and the metrical geometry of operator space, and 'The Special Canonical Group' having obtained the continuous q, p degree of freedom as the limit of finite measurement algebras):

(1) Dynamics-measurement consistency. Construct a detailed formulation of quantum dynamics - within standard quantum theory or a principled extension of it (an extension counting as principled only when its modification is fixed by an independently stated assumption or predicts a testable deviation from unitary evolution) - whose description of the interactions that constitute measurement reproduces the symbolic characterization derived at the kinematic stage of Schwinger ('The Algebra of Microscopic Measurement', Proceedings of the National Academy of Sciences 45(10), 1542-1553 (1959), DOI: 10.1073/pnas.45.10.1542, https://doi.org/10.1073/pnas.45.10.1542). In that characterization: M(a′,a′′)M(a',a'') denotes the selective measurement that accepts systems found in the state a′′a'' and produces them in the state a′a'; the symbols multiply noncommutatively (products are read right to left, as measurements performed in succession) according to M(a′,a′′)M(a′′′,a′′′′)=δ(a′′,a′′′)M(a′,a′′′′)M(a',a'')M(a''',a'''') = \delta(a'',a''')M(a',a''''), where δ(a′′,a′′′)\delta(a'',a''') is the Kronecker symbol, equal to 1 when the two values coincide and 0 otherwise; the transformation function ⟨a′∣b′⟩\langle a'|b'\rangle is the complex number relating two incompatible complete descriptions aa and bb of the same system, with the composition property ∑b′⟨a′∣b′⟩⟨b′∣c′⟩=⟨a′∣c′⟩\sum_{b'}\langle a'|b'\rangle\langle b'|c'\rangle = \langle a'|c'\rangle for any third complete description cc; the probability law assigns to the pair of states a′a', b′b' the real, nonnegative number p(a′,b′)=∣⟨a′∣b′⟩∣2p(a',b') = |\langle a'|b'\rangle|^2, which is symmetric in the two states, p(a′,b′)=p(b′,a′)p(a',b') = p(b',a'), and additive in the sense that a measurement that does not distinguish between several values yields the sum of the probabilities of those values; and the nonselective measurement, in which the apparatus functions but selects nothing, is represented by the sum of the selective symbols with independent randomly distributed phases - the algebraic expression of the uncontrollable disturbance produced by measurement. The construction must exhibit a mechanism by which the microscopic effects of a system-apparatus measurement interaction are amplified to the level of macroscopic observation, producing a macroscopically distinguishable, stable apparatus record; the probability assignment and the selective measurement state change must both be derived from the dynamics rather than invoked as inputs, and every other element of the symbolic characterization that is invoked as an input must be explicitly identified rather than tacitly assumed.

(2) The complex-number structure. Determine the physical significance of the appearance of complex numbers in the measurement algebra - in particular whether complex conjugation can be interpreted, as Schwinger conjectured in 1958 and repeated in the footnote of the 1959 paper, as the change of description accompanying the substitution of an antimatter apparatus for a matter apparatus (or conversely), that is, whether the complex-number structure of quantum mechanics reflects the fundamental matter-antimatter duality. The matter-antimatter reading requires a concrete antilinear (charge-conjugation-like) operation on the measurement symbols that implements the substitution of an antimatter apparatus for a matter apparatus, together with at least one structural or observable consequence that would confirm or refute Schwinger's conjecture; a negative finding must exclude any such operation, rather than rest on the fact that complex numbers already follow from the positivity of the probability law. If the answer is affirmative, indicate how the reconstruction of quantum field theory that Schwinger asserted to be possible on this basis, and deferred to another occasion, is to proceed; this further component is conditional on an affirmative answer.

The two components are the distinct surviving open items of Schwinger's program: the dynamics consistency requirement stated in the closing paragraph of the 1959 paper, and the complex-number question deferred in its footnote 2 and in the 1958 TCP paper.

References

  1. Julian Schwinger, 'THE ALGEBRA OF MICROSCOPIC MEASUREMENT', Proceedings of the National Academy of Sciences of the United States of America 45(10), 1542-1553 (1959). DOI: 10.1073/pnas.45.10.1542, https://doi.org/10.1073/pnas.45.10.1542
  2. Julian Schwinger, 'SPIN, STATISTICS, AND THE TCP THEOREM', Proceedings of the National Academy of Sciences of the United States of America 44(2), 223-228 (1958). DOI: 10.1073/pnas.44.2.223, https://doi.org/10.1073/pnas.44.2.223
  3. Julian Schwinger, 'THE GEOMETRY OF QUANTUM STATES', Proceedings of the National Academy of Sciences of the United States of America 46(2), 257-265 (1960). DOI: 10.1073/pnas.46.2.257, https://doi.org/10.1073/pnas.46.2.257
  4. Julian Schwinger, 'UNITARY OPERATOR BASES', Proceedings of the National Academy of Sciences of the United States of America 46(4), 570-579 (1960). DOI: 10.1073/pnas.46.4.570, https://doi.org/10.1073/pnas.46.4.570
  5. Julian Schwinger, 'THE SPECIAL CANONICAL GROUP', Proceedings of the National Academy of Sciences of the United States of America 46(10), 1401-1415 (1960). DOI: 10.1073/pnas.46.10.1401, https://doi.org/10.1073/pnas.46.10.1401
  6. Julian Schwinger, 'Quantum Kinematics and Dynamics', CRC Press (2018 reissue; chapters: The Algebra of Measurement; The Geometry of States; The Dynamical Principle; The Special Canonical Group; Canonical Transformations; Groups of Transformations; Canonical Transformation Functions; Green's Functions; Some Applications and Further Developments). DOI: 10.1201/9780429497933, https://doi.org/10.1201/9780429497933
  7. Julian Schwinger, 'Quantum Mechanics: Symbolism of Atomic Measurements' (edited by Berthold-Georg Englert), Springer, Berlin Heidelberg (2001), ISBN 978-3-540-41408-7. DOI: 10.1007/978-3-662-04589-3, https://doi.org/10.1007/978-3-662-04589-3
  8. Gordon Reece, 'The theory of measurement in quantum mechanics', International Journal of Theoretical Physics 7, 81-116 (1973). DOI: 10.1007/BF00713794, https://doi.org/10.1007/BF00713794
  9. M. D. Srinivas, 'Collapse postulate for observables with continuous spectra', Communications in Mathematical Physics 71, 131-158 (1980). DOI: 10.1007/BF01197917, https://doi.org/10.1007/BF01197917
  10. Z. Maki, 'An Algebraic Approach to the Quantum Theory of Measurements', Progress of Theoretical Physics 84(4), 574-583 (1990). DOI: 10.1143/PTP/84.4.574, https://doi.org/10.1143/PTP/84.4.574
  11. L. P. Horwitz, 'Schwinger algebra for quaternionic quantum mechanics', Foundations of Physics 27, 1011-1034 (1997). DOI: 10.1007/BF02551150, https://doi.org/10.1007/BF02551150 (also arXiv:hep-th/9702080, https://arxiv.org/abs/hep-th/9702080)
  12. P. Watson and A. J. Bracken, 'Quantum Phase Space from Schwinger's Measurement Algebra', Foundations of Physics 44, 762-780 (2014). DOI: 10.1007/s10701-014-9813-1, https://doi.org/10.1007/s10701-014-9813-1
  13. F. M. Ciaglia, A. Ibort, G. Marmo, 'A gentle introduction to Schwinger's formulation of quantum mechanics: The groupoid picture', Modern Physics Letters A 33, 1850122 (2018). DOI: 10.1142/S0217732318501225, https://doi.org/10.1142/S0217732318501225
  14. F. M. Ciaglia, A. Ibort, G. Marmo, 'Schwinger's picture of quantum mechanics I: Groupoids', International Journal of Geometric Methods in Modern Physics 16, 1950119 (2019). DOI: 10.1142/S0219887819501196, https://doi.org/10.1142/S0219887819501196
  15. Wojciech Hubert Zurek, 'Decoherence, einselection, and the quantum origins of the classical', Reviews of Modern Physics 75, 715-775 (2003). DOI: 10.1103/RevModPhys.75.715, https://doi.org/10.1103/RevModPhys.75.715
  16. Angelo Bassi, Kinjalk Lochan, Seema Satin, Tejinder P. Singh, Hendrik Ulbricht, 'Models of wave-function collapse, underlying theories, and experimental tests', Reviews of Modern Physics 85, 471-527 (2013). DOI: 10.1103/RevModPhys.85.471, https://doi.org/10.1103/RevModPhys.85.471
  17. Marc-Olivier Renou, David Trillo, Mirjam Weilenmann, Thinh P. Le, Armin Tavakoli, Nicolas Gisin, Antonio Acín, Miguel Navascués, 'Quantum theory based on real numbers can be experimentally falsified', Nature 600, 625-629 (2021). DOI: 10.1038/s41586-021-04160-4, https://doi.org/10.1038/s41586-021-04160-4
  18. Jürg Fröhlich, Alessandro Pizzo, 'The Time-Evolution of States in Quantum Mechanics according to the ETH-Approach', Communications in Mathematical Physics 389, 1673-1715 (2021). DOI: 10.1007/s00220-021-04225-5, https://doi.org/10.1007/s00220-021-04225-5