Abstract
While there exists a vast literature devoted to logical analysis of physical theories the literature on intentheory relations is almost nonexistent. True, intertheory relations are touched upon by many authors, both scientists and philosophers of science, and the interest in intertheory relations is rising rapidly. Yet the systematic study of them has hardly begun, a notable exception being the work of Tisza1 which, however, is complementary to rather than concurrent with the present work.
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Abbreviations
- c-theory:
-
Einstein’s Special Theory
- c-K-theory:
-
Einstein’s General Theory
- CPM:
-
Classical Point Mechanics
- h-:
-
quantum, quantum mechanical
- HS:
-
Hilbert Space
- ITR:
-
intertheory relation [al]
- MF:
-
mathematical formalism (of a physical theory)
- MFS :
-
mathematical substructure (of a physical theory)
- MFS :
-
mathematical superstructure (of a physical theory)
- model1 :
-
model in the ordinary sense (if B is a model1 of A, B is on a higher level of abstraction than A)
- model2 :
-
model in the sense of Mathematical Logic (if B is a model2, of A, B is on a lower level of abstraction than A)
- o-:
-
classical, not containing any universal constant
- PM:
-
Point Mechanics, mechanics of mass points
- PI:
-
physical interpretation (of an MF)
- prob1 :
-
distribution probability
- prob2 :
-
transition probability
- QFT:
-
Quantum Field Theory
- QM:
-
Quantum Mechanics, quantum mechanical
- RD(T):
-
region of definition of T
- RV(T):
-
range of validity of T
- S f :
-
classical system of f degrees of freedom
- T:
-
(physical) theory
References
Tisza, L., The Conceptual Structure of Physics’, Reviews of Modern Physics 35 (1963) 151–85.
The subject matter of Mathematical Logic, a subject taught at some universities, is the logic of mathematics rather than mathematical logic which is logic treated in a mathematical way or, equivalently, mathematics admitting logical models2.
Strauss, M., ‘Mathematics as Logical Syntax — A Method to Formalize the Language of a Physical Theory’, Journal of Unified Science (Erkenntnis) 7 (1938).
Strauss, M., ‘Zur Begründung der statistischen Transformationstheorie der Quantenphysik’, Sitzungsberichte der Berliner Akademie der Wissenschaften, Physikalischmathematische Klasse 27 (1936) 382–98; ‘Grundlagen der modernen Physik’, in Mikrokosmos-Makrokosmos Vol. II (ed. by H. Ley and R. Loether ), Berlin 1967.
With c→ ∞ the Minkowski invariant (ΔS)2 = c2(ΔT)2 — (ΔL)2 does not split up into the two invariants of Newtonian space-time but becomcs infinite.
For a formulation of c-kinematics with redundant parameters cf. M. Strauss, ‘On a Generalized Lorent Transformation’, Annalen der Physik 16 (1965) 105–13.
According to present terminology the two ‘systems’ are models1 rather than theories.
Landé, A., New Foundations of Quantum Mechanics, Cambridge 1965.
Cf. ref. 4.
Strauss, M., ‘The Lorentz Group: Axiomatics — Generalizations — Alternatives’, Wissenschaftliche Zeitschrift der Friedrich-Schiller-Universität Jena, Mathematisch-Naturwissenschaftliche Reihe 15 (1966) 109–18, and ref. 4, Part III.
Cf., e.g., H. Grad, ‘Levels of Description in Statistical Mechanics and Thermodynamics’, and Edwin T. Javnes, ‘Foundations of Probability Theory and Statistical Mechanics’, in Delaware Seminar in the Foundations of Physics (ed. by M. Bunge ), Berlin-Heidelberg-New York 1967.
Havas, P., ‘Foundation Problems in General Relativity’, in Delaware Seminar in the Foundations of Physics (ed. by M. Bunge ), Berlin-Heidelberg-New York 1967.
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Bateman, H.: 1909/10, ‘The Transformation of the Electrodynamic Equations’, Proc. London Math. Soc., ii. ser., 8, 223–264.
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Cassirer, E. (ed.): 1904, G. W. Leibniz — Hauptschriften zur Grundlegung der Philosophie, Leipzig, pp. 242–245.
Cunningham, E. : 1909/10, ‘The Principle of Relativity in Electrodynamics and an Extension thereof’, Proc. London Math. Soc., ii. ser., 8, 77–98.
Dautcourt, G.: 1964, ‘Die Newtonische Gravitationsthcorie als strenger Grenzfall der Allgemeinen Relativitätstheorie’, Acta Physica Polonica 25, 637–647.
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Herneck.F.: 1966, ‘Die Beziehungen zwischen Einstein und Mach, dokumentarisch dargestellt’, Wiss. Z. Friedrich-Schiller-Univ. Jena. Math.-Nat. R., Heft 1,15,1–14.
Hönl, H.: 1966a, ‘Zur Geschichte des Machschen Prinzips’, Wiss. Z. Friedrich-Schiller-Univ. Jena, Math.-Nat. R., 15, 25–36.
Hönl, EL: 1966b, ‘Das Machsche Prinzip und seine Beziehung zur Gravitationstheorie Einsteins’, in Einstein-Symposium (ed. by H.-J. Treder ), Berlin, pp. 238–278.
Kretschmann, E.: 1917, ‘Über den physikalischen Sinn der Relativitätspostulate’, Ann. d. Phys. 53, 575.
Leibniz, G.W.: 1691–95, Briefwechsel mit Huygens, s. Cassirer (1904).
Lenin, W.I.: 1909, Materialismus und Empiriokritizismus, Moskau. (In Russian.)
Lubkin, E.: 1961, Frames and Lorentz Invariance in General Relativity, Univ. of California, UCRL-9668 Internal, April 19.
Mach, E.: 1883, Die Mechanik in ihrer Entwicklung, historisch-kritisch dargestellt, Leipzig.
Mach,E.: 1917, Erkenntnis und Irrtum, 3rd ed., Leipzig.
Magic, W.F.: 1935, A Source Book in Physics, New York-London.
Milne, E.A: 1948, Kinematic Relativity, Oxford.
Noether, E.: 1918, ‘Invariante Variationsprobleme’, Göttinger Nachr., 235–257.
Pirani, F.A.E.: 1957, Tetrad Formulation of General Relativity Theory’, Bull. Acad. Polon. Sci. 5, 143–147.
Planck, M.: 1910, ‘Zur Machschen Theorie der physikalischen Erkenntnis’, Phys. Z. 11, 1186.
Reichenbach,H.: 1924, ‘Die Bewegungslehre bei Newton, Leibniz und Huygens’, Kantstudien 29, 416.
Schlipp, P.A. (ed.): 1949, Albert Einstein: Philosopher-Scientist, Evanston, Ill.
Strauss, M.: 1938, ‘Mathematics as Logical Syntax — A Method to Formalize the Language of a Physical Theory’, J. of Unified Science (Erkenntnis) 7.
Strauss, M.: 1957/58, ‘Grundlagen der Kinematik, I — Die Lösungen des kinematischen Transformationsproblcms’, Wiss. Z. Humboldt-Umv.-Berlin, Math.-Nat. R., 7, 609–616.
Strauss, M.: 1960, ‘Max Planck und die Entstehung der Quantentheorie’, in Forschen und Wirken, Festschrift zur 150-Jahr-Feier d. Humboldt-Univ. Berlin, I, Berlin, pp.
Strauss, M.: 1965, ‘On the Voigt-Palacios-Gordon Transformation and the Kinematics Implied by it’, Nuovo Cim. (X) 39, 658–666.
Strauss, M.: 1966, The Lorentz Group: Axiomatics — Generalizations — Alternatives’, Wiss. Z. Friedrich-Schiller- Univ. Jena, Math.-Nat. R. 15, 109–118.
Strauss, M.: 1967a, ‘Zur Logik der Begriffe “Inertialsystem” und “Masse”’, in Ernst- Mach-Symposium 1966 in Freiburg (in press).
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Treder, H.-J.: 1966, ‘Lorentz-Gruppe, Einstein-Gruppe und Raum-Zeit-Struktur’, in Einstein-Symposium (ed. by H.-J. Treder ), Berlin, pp. 57–75.
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References
The expression ‘Machsches Prinzip’ is due to Einstein (1918) and denotes an interpretation of Mach’s comment on Newtonian mechanics in terms of Riemannian field theory: cf. Ref. 35.
Cf. e.g. Lenin (1909) and Planck (1910).
Cf. the review article on Mach’s Erkenntnis und Irrtum, by F. Jodl, republished as Appendix to this work on the direction of Mach. - Mach (1917), pp. 464–470.
Hönl (1966a, b), Gürsey (1963), Wheeler (1963b).
These names are used for what is commonly called ‘Special Theory of Relativity’ and ‘General Theory of Relativity’, respectively. These and similar names involving ‘relativity’ are misleading, as is now generally recognized. The novel name ‘(Einstein’s) Theory of Gravitation’ advocated by Fock does not show that the General Theory contains more than a theory of gravitation: the General Theory generalizes the Special Theory in such a way that a theory of gravitation is included.
This is no blemish on the theories’ inventor: the history of science knows no instance of a physical theory correctly understood by its author. The correct meaning of a new fundamental physical theory only emerges in a long and difficult process of logico-mathematical analysis and practical applications.
Noether (1918).
For proofs cf. Appendix.
Cf. Strauss (1938).
Cf. Strauss (1966).
Cf. M. Strauss (1966), section 4.
Except when the spacetime degenerates into a direct product of time and 3-space as in the Newtonian case. In general the metric of a 3-space t=const, is given by γαβ=gαβ+g0αg0β/g00 (α, β = 1, 2, 3). (In the General Theory this defines the metric of ‘light geometry’.)
From the mathematical point of view the two kinds of transformations refer to dual spaces related by the quantities (1) \({h_\alpha }\alpha = \frac{{\partial {X^\alpha }}}{{\partial {x^\alpha }}}\) and (2) \({h_\alpha }\alpha = \frac{{\partial {x^\alpha }}}{{\partial {X^\alpha }}}\), where the Greek indices refer to coordinate systems and the Latin indices to frames. To every ‘space tensor’ (tensor under coordinate transformations) T χλ … αβ … there exists a ‘frame tensor’ (tensor under frame transformations) T kl … ab …and vice versa according to (3) \({T_{kl \ldots }}^{ab \ldots } = {h_\alpha }^a{h_\beta }^b \ldots {h_k}^x{h_l}^\lambda \ldots {T_{x\lambda \ldots }}^{\alpha \beta \ldots }\) (4) \({T_{x\lambda \ldots }}^{\alpha \beta \ldots } = {h_a}^\alpha {h_b}^\beta \ldots {h_x}^k{h_\lambda }^l \ldots {T_{kl}}^{ab \ldots } \cdot \) In particular, the metrical tensors n ab and g αβ defined by (5) \({\left( {dS} \right)^2} = {\eta _{ab}}d{X^a}d{X^b} = {g_{\alpha \beta }}\left( x \right)d{x^\alpha }d{x^\beta }\) are related by (6) \({\eta _{ab}} = {h_a}^\alpha {h_b}^\beta {g_{\alpha \beta }}\left( { = \pm 1} \right)\) (7) \({g_{\alpha b}} = {h_\alpha }^a{h_\beta }^b{\eta _{ab}}\left( { = \pm 1} \right)\). Quantities like (5) that are invariant under both coordinate and frame transformations are to be called proper invariants. The frame tensors (3) have direct physical meaning since their values do not depend on the arbitrary choice of the coordinate system; they are the quantities that may be measured in a local frame. The space tensors (4) have no direct physical meaning; but of course space tensor equations have, since T χλ … αβ …≡ 0 implies T kl … ab… ≡ 0. For more comprehensive treatment cf. Lubkin (1961), Treder (1966), Pirani (1957).
Our ‘T’ means frame time (extended time), whereas our ‘t’ is a general time coordinate.
The decision what interval is to be considered ‘infinitesimal’ depends on the curvature R at the given world point: the condition is |dS|:R −2≪1.-The condition |dS| ≪ 1 would have no meaning since |dS| is a dimensional quantity (length). Thus, the existence of a second dimensional invariant R is necessary for a consistent and meaningful physical interpretation of the General Theory.
These technical terms are misleading since they suggest motion between the frames.
Any set γ may be said to be the direct product of sets γ1 and γ2 if and only if any element of γ is a pair of elements, one from γ1 and one from γ2, and vice versa.
This group Ki is given by the frame transformations (IEqua1) with (group conditions) f ( ii )(T)≡0 (identity) f ( ik )(T) = −f( ki )(T) (inverse) f ( il )(T) = f ( ik )(T) + f ( kl )(T) (composition). If f ( ik )(T)=v ( ik ) T, one obtains the familiar (proper) Galilei group. In general, f ( ik )(T) contains all time derivatives, as may be seen from serial expansion. Thus the (proper) Galilei group is but a small subgroup of K1.
The group K2 is given by the frame transformation (IEqua) with (group conditions) ω( i ) b ( i ) a =δba(identity) ω( i ) b ( k ) a × ω( k ) c ( i b=δca(inverse) ω( i ) b ( i ) a =ω( k ) b ( i ) a× ω( l ) c ( k) b(composition).
A simply connected n-dimensional space of maximal symmetry admits N = n(n + 1)/2 independent symmetry operations; this gives N = 10 for n = 4. But Newtonian space-time is not a simply connected 4-dimensional space. (An element of Newtonian space-time is not a point but a pair of points.)
This was first shown by Cunningham (1910) and Bateman (1910). Cf. also Cashmore (1963) and Strauss (1966), p. 110.
This allows for additional conservation laws (such as that for electric charge).
Cf. Alexander (1956), Leibniz (1691–95).
Not vindicated is Newton’s conception of ‘absolute space’. However, this conception does not play any role in Newtonian mechanics: ‘absolute acceleration’ means acceleration with respect to any one of the inertial frames. For this reason it is usually held to be a metaphysical construct. Yet ‘absolute space’ as conceived by Newton is an inconsistent concept: on the one hand it is supposed to be homogeneous and isotropic like the ‘relative spaces’ of our experience, on the other hand a displacement in absolute space is supposed to correspond to a real or fictitious process, which implies inhomogencity. Thus, a mathematical model of Newton’s ‘absolute space’ is impossible. To turn Newton’s conception of ‘absolute space’ into a consistent notion it has to be reinterpreted to mean ‘uniquely determined preferential frame’. (The Maxwell equations were once thought to define such a frame.)
Cf. Dautcourt (1964) and the literature quoted there.
Mach (1883), quoted after Heller (1964), p. 47.
Mach (1883), quoted after Heller (1964), p. 32.
Schilpp (1949), p. 52.
Einstein (1916b), quoted after Heller (1964), p. 156.
Milne (1948).
Strauss (1957/58); cf. also Strauss (1966).
Mach (1883), quoted after Heller (1964), p. 34.
Mach (1883), quoted after Heller (1964), p. 39.
The exclusion of conservative forces depending on velocity by Newtonian mechanics was one of the reasons for Maxwell to reject the mechanical theory of Weber. He wrote: “(2) The mechanical difficulties, however, which are involved in the assumption of particles acting at a distance with forces which depend on their velocities are such as to prevent we from considering this [Weber’s] theory as an ultimate one. …”. (quoted after Magie (1935), p. 529).
Machsches Prinzip: Das G-Feld [gµv-field, M.S.] ist restlos durch die Massen der Körper bestimmt. Da Masse und Energie nach den Ergebnissen der Speziellen Relativitätstheorie das gleiche sind und die Energie formal durch den symmetrischen Encrgietcnsor (T µv) beschrieben wird, so besagt dies, daß das G-Feld durch den Energietensor der Materie bedingt und bestimmt sei. (Einstein, 1918 ).
Hönl (1966a, b).
Even in his 1918 paper, quoted in Ref. 35, Einstein calls his ‘Mach Principle’ a generalization of “die Machsche Forderung, daß die Trägheit auf eine Wechselwirkung der Körper zurückgeführt werden müsse”.
For a somewhat fuller discussion of this point cf. Strauss (1967a).
For the genesis of Planck’s quantum theory cf. Strauss (1960).
Einstein (1916a) thought that the postulate of general covariance “dem Raum und der Zeit den letzten Rest physikalischer Gegenständlichkeit nehmen würde” and that only ‘coincidences’, i.e., the points of intersection of world lines, would remain as objective facts.
References
Cf. ref. 4 Part I.
In fact, any PI different from the standard one would give a different physical theory. See also ref. 4.
Cf. ref. 8, Part I.
Many interpretations, offered as mere reconceptualizations, are in fact non-standard PIs. A typical example is Bohm’s ‘causal interpretation’. In spite of this author’s contention to the contrary, his ‘interpretation’ gives entirely different results. For instance, it predicts that a hydrogen atom has a magnetic moment even in the ground state, in contradiction to the standard PI.
This relation can be established in infinitely many ways, each one characterized by a phase factor exp (ia) common to all state vectors. This fact is often described by saying that the state vector is only defined ‘up to an arbitrary phase factor’. This is utterly misleading as it suggests a one-many relation between states and state vectors.
We omit the normalizing factor (Tr P1)-1.
The set theoretical distinction between cardinal numbers is the coarsest one possible. If we omit one |>k from a complete set {|>i} the resulting set {|>i}i≠ k has the same set theoretical cardinal number as the complete set though it is no longer complete. Thus, in HS we can distinguish between co (meaning the cardinal number of a complete set) and ∞ − 1.
For detailed discussion and proofs cf. J.-M. Lévy-Leblond, ‘Galilei Group and Nonrelativistic Quantum Mechanics’, J. Math. Phys. 4 (1963) 776–88.
A great deal of confusion has been generated by the use of statistical terms for which the referent, i.e. the ensemble, is not specified. Thus, the expression (a) \(\bar A\mathop = \limits_{df} \left\langle {\left| A \right|} \right\rangle = Tr{P_{\left. | \right\rangle }}A = {\sum {\left| {{{\left\langle | \right\rangle }_i}} \right|} ^2}{a_i}\) is usually spoken of as ‘mean value’ or — worse still — as ‘expectation value’ of A for (or in) the state. This is nonsense if taken verbally since neither a mean value nor an expectation value is defined for a single system; it is wrong if the ensemble to which the mean value refers is taken to be the uniform ensemble (‘pure case’) of systems all in state: in this ensemble A has no mean value at all since none of the ensemble members is in a state where A has any definite value. In fact, the expression (a) is the mean value of (the eigenvalues of) A in the nonuniform ensemble (‘mixture’) in which the eigenstate of A occurs with the relative frequency (b) \({h_i} = {\left| {{{\left\langle | \right\rangle }_i}} \right|^2} = pro{b_2}\mathop {\left( {\left| {\rangle \to \left| {{\rangle _i}} \right.} \right.} \right)}\limits^A \) To call expression (a) the ‘expectation value of A’ for a ‘measurement’ of A is also wrong since in general the value of expression (a) will be different from any eigenvalue of A and hence the ‘expectation’ of finding value (a) by a ‘measurement’ of A will be exactly zero!
Cf. E. Schrödinger, Statistical Thermodynamics, Cambridge 1946.
Cf. M. Strauss, ‘Entwicklungsgesetze and Perspectiven der Physik’, Monatsber. Dtsch. Akad. bliss. Berlin 9 (1967) 538–47.
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Strauss, M. (1970). Intertheory Relations. In: Weingartner, P., Zecha, G. (eds) Induction, Physics and Ethics. Synthese Library, vol 31. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-3305-3_12
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