Abstract
QM is notoriously associated with a certain ‘strangeness’ or ‘weirdness’ (e.g. Rosenblum and Kuttner 2011, p. 4; Davies 2004, p. 11) which stems, in the first place, from the divergence of the phenomena that it describes and predicts from our pre-quantum expectations. By ‘phenomenon’ we here mean, for practical reasons, something along the lines of Bogen and Woodward (1988, pp. 305–306), according to whom the phenomenon is rather what the theory predicts, which may not even be observable, whereas the data are the observables that serve as evidence for phenomena.
…if one is not shocked about the quantum theory at first, one cannot possibly have understood it.
—Attributed to N. Bohr by Heisenberg (cf. 1969, p. 241)
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Notes
- 1.
Caveat: Parts of our exposition of the historical details will be quite ‘whig’ in the sense of Butterfield (1931), meaning that one “studies the past with reference to the present[…].” (ibid., section 2)
- 2.
- 3.
More specifically, the classical expression for intensity can be calculated from the average magnitude of the Poynting-vector \(\boldsymbol {S} = \frac {1}{\mu _{0}} \boldsymbol {E} \times \boldsymbol {B}\) as \(\langle S \rangle = \frac {1}{\mu _{0}}\langle EB \rangle = \frac {1}{c\mu _{0}}\langle E^{2} \rangle \), with μ 0 the vacuum permeability, c the velocity of light, and E and B the magnitudes of the electric and magnetic field vectors (e.g. Walker et al. 2012, p. 981).
- 4.
The frequency has the dimension 1/time, the wavelength is a length, so the product has the dimension length/time, which is that of a velocity. Strictly speaking, what we appeal to above is the so called phase velocity which may in fact exceed the speed of light, c, when multiple waves travel together as a ‘packet’, a narrow lump of oscillations. So it is basically the ‘false kind of velocity’. The velocity of interest is the group velocity
, which, however, in the present case of a single wave coincides with the phase velocity, so that no harm comes from the simplification. Note that no energy or information can be transported with phase velocity: pictorially it describes how fast the ‘ripples’ in the packet propagate, but the ripples ‘diminish’ while approaching the boundary of the packet, and so no energy or information is transmitted with a speed > c (e.g. Griffiths 1995, pp. 47–48; Griffiths 1999, p. 399 for details, examples, and illustrations). This is, of course, important for consistency with relativity. - 5.
E.g. Basdevant and Dalibard (2002, p. 343) for the details.
- 6.
The actual experiment described by Greenstein and Zajonc (2006, pp. 103) in fact involves multiple atoms. Therein detectors are used, however, that accept only single photons at a time, so that Greenstein and Zanjonc deem the arrangement “very close to an ideal experiment, in which we work with one atom and one photon at a time.” (ibid.)
- 7.
Recall that p ⋅x is an inner product which computes p xx + p yy + p zz, which is why E ⋅ t can be meaningfully subtracted from it.
- 8.
Note that \(\frac {(\hat {\boldsymbol {p}})^{2}}{2m}= \frac {(-i \hbar \nabla )^{2}}{2m} = -\frac {\hbar ^{2}}{2m} \Delta \).
- 9.
Terminology may be confusing here, since the Greek δύναμις, from which the term derives, actually means ‘power, ability’ (cf. Perschbacher 1990, p. 108). The modern use of the term can be connected to Newtonian physics, where considerations of forces give rise to the differential equations describing the time evolution of systems. The introduction of term into physics is typically traced back to the dynamism of Leibniz (cf. Bernstein 1981, p. 97).
- 10.
Cf. Joos et al. (2003, p. 7).
- 11.
Equation (2.12) can also be straightforwardly proven, for instance by appeal to a convergence generating factor.
- 12.
We here take the widely used notion of a ‘system’, which Bell (1990a, p. 34) complained, should be purged from a physically precise theory altogether, to be simply ontologically as non-committal as the term ‘entity’ in philosophy, and hence without any general implication of an involvement of, say, parts and wholes.
- 13.
The proportionality can be given in the form \(\boldsymbol {\mu } = g_{s}\frac {\mu _{s}}{\hbar }\boldsymbol {s}\) where g s varies with the particle sort, and \(\mu _{s}=\frac {q\hbar }{2m_{0}c}\) is called a magneton, with ħ Planck’s reduced constant, q the particle’s charge, c the speed of light, and m 0 its rest mass (cf. Haken and Wolf 1996, p. 188; Mayer-Kuckuk 2002, p. 58).
- 14.
If terminology is starting to sound unfamiliar, please at any rate consult Appendix A.
- 15.
In terms of units or physical dimensions we can thus see how the spin is still ‘reminiscent’ of an angular momentum; the dimension of ħ is energy × time which is equal to \(\frac {\text{mass}\times \text{lenght}^{2}}{\text{time}}\), the dimension of an (intrinsic) angular momentum I ω, with I the moment of inertia.
- 16.
Of course there is a debate on the existence of so called paraparticles which obey a third kind of statistics (cf. Massimi 2005, p. 154 ff.). The connection between spin and statistics, moreover, can only be thoroughly established in the formalism of QFT, and it here appears merely as an inductive generalization. But we will not pursue either of these issues any further in this book.
- 17.
This is the notation also used by Redhead (1987, p. 8).
- 18.
Note that we are effectively avoiding such things as the ‘big vs. many’-debate by simply stipulating that what is conditioned on has probabilities (cf. Wroński 2014, p. 45ff.). For convenience, we will do so throughout this book.
- 19.
‘Function’ is meant here in the neutral, set-theoretic sense, not in the specific sense of calculus. If you prefer this, you can replace it in thought by the more neutral ‘map’, which also covers measures (cf. Appendix A for details).
- 20.
Again, depending on the interpretation of p, it must measure the probability that variable \( \underline {X}\) takes on a certain value for some individual S ∈ D, not a specific one.
- 21.
As should be clear by now, joint probabilities are a delicate matter in QM; for what, say, is the joint probability \(p( \underline {s_{x}}(S)=+\frac {\hbar }{2}, \underline {s_{z}}(S)=+\frac {\hbar }{2})\), understood as ‘equal time’? There does not seem to be an answer; both measurement procedures mutually exclude each other. That the matter is ‘even more delicate’ than mere limitations of joint measurability will become obvious in the following.
- 22.
Taking the squared modulus |〈⋅|⋅〉|2 not only ensures real values, but also that the function Pr satisfies the first Kolmogorov axiom (cf. Appendix A) Pr(a) ≥ 0 for all a in the domain of Pr.
- 23.
Cf. Joos et al. (2003, p. 7).
- 24.
That this implies that no ‘genuine’ superpositions of the form |ξ〉 = α|ψ〉 + β|ϕ〉 can exist becomes clear by appeal to the density operator, thoroughly introduced later. With
, one has 
which would equally result from 
with 
the density operator of a proper mixture (cf. 2.1.5), so that a coherent superposition of |ψ〉 and |ϕ〉 is indistinguishable from a proper mixture (cf. Giulini 2009, p. 773). - 25.
κίνεσις is (ancient) Greek for ‘motion’ (cf. Perschbacher 1990, p. 240), so the term ‘kinematics’ is again related to Newtonian physics and the fact that therein states are obtained by solving an equation of motion. It (or rather the french cinématique) was first suggested by Ampère (1838) in his Essai sur la philosophie des sciences, “for a field of mechanics that would be concerned with motion independent of its causes.” (Koetsier 1994, p. 994)
- 26.
In fact, the commutator will here give back another Pauli matrix, up to a multiple of i; the commutation relations can be summarized in the form \([\hat {\sigma }_{k}, \hat {\sigma }_{\ell }] = i2\sum _{m=1}^{3}\epsilon _{k\ell m}\hat {\sigma }_{m}\) with
replaced by 
, and where 𝜖 is the so called Levi-Civita symbol which gives back 1 if k, ℓ, and m are cyclical permutations of 1, 2, 3, − 1 for anti-cyclical ones, and 0 if two of the numbers are identical. In virtue of this property, the Pauli matrices form an abstract commutator algebra (a Lie algebra), as do other types of angular momentum in QM (e.g. Schwindt 2013, pp. 197 ff. and 256 ff.). - 27.
Still, it is clear that this point is actually of greater concern and has the potential to raise controversy. Moreover, note that not every self-adjoint operator can correspond to an observable (cf. Footnote 24) whereas the converse might just be the case (e.g. d’Espagnat 1995, p. 98).
- 28.
The square ensures that all the deviations are counted positively.
- 29.
Again we stick to the standard QM-terminology here nevertheless, as was the case with ‘observable’ or ‘system’.
- 30.
For asserting ‘completeness’ it is here assumed to be sufficient that for N-tuples of eigenvalues (λ 1, …λ N) of the operators in a commuting set
there are simultaneous eigenvectors \({\left\vert \psi _{\lambda _{1},\ldots \lambda _{N}}\right\rangle}\), such that the set of these simultaneous eigenvectors span the space \(\mathbb {H}\) (cf. Ruetsche 2011, p. 200). - 31.
Here \(\hat {p}_{x}\) may be represented as
on a space of functions of position, ψ(x), and \(\hat {x}\) merely multiplies the latter by x. Then one sees immediately that 
. Note that in a space of momentum-dependent functions, \(\tilde {\psi }(p)\), \(\hat {x}\) would be a derivative w.r.t. p
x, and \(\hat {p}_{x}\) a multiplication-operator. - 32.
With an eye on Appendix A and the brief discussion of rigged Hilbert spaces therein, this means, strictly speaking, that \({\langle \varphi \hat {\boldsymbol {x}}|\boldsymbol {x}_{0}\rangle}=\boldsymbol {x}_{0}{\langle \varphi |\boldsymbol {x}_{0}\rangle}\) for φ ∈ Φ (cf. de la Madrid 2005, p. 302).
- 33.
A more thorough definition of the position operator is possible in terms of operator valued measures (cf. Appendix A and Heinosaari and Ziman 2012, pp. 128–131).
- 34.
Weinberg (2013, pp. 14–21) gives a nice overview of some matrix mechanics; so the interested reader may be referred there.
- 35.
For general discussion cf. Ruetsche (2011, chapters 2 and 3); for a statement of the theorem cf. p. 41 therein; and for proofs cf. the references therein.
- 36.
Note, however, that atomic physics already constitutes an example where fully analytic treatments without approximations are rare; strictly speaking this is only possible for the single-electron problem, i.e. for the hydrogen atom.
- 37.
We here closely follow Basdevant and Dalibard (2002, p. 80 ff. and p. 120 ff.).
- 38.
The more general case of t 0 ≠ 0 would require \(\hat {U}(t_{0};t)=e^{-\frac {i}{\hbar }\hat {H}(t-t_{0})}\). If we then let t − t 0 = 𝜖, we can write \({\vert \psi (t_{0}+\epsilon )\rangle}=e^{-\frac {i}{\hbar }\hat {H}\epsilon }{\vert \psi (t_{0})\rangle} = (1-\frac {i}{\hbar }\hat {H}\epsilon +\mathbb {O}(\epsilon ^{2})){\vert \psi (t_{0})\rangle} \Leftrightarrow i\hbar \frac {{\vert \psi (t_{0}+\epsilon )\rangle}-{\vert \psi (t)\rangle}}{\epsilon } = \hat {H}{\vert \psi (t_{0})\rangle} + \mathbb {O}(\epsilon ){\vert \psi (t_{0})\rangle}\), where \(\mathbb {O}(\epsilon ^{k})\) means ‘terms of order 𝜖 k’ (i.e. wherein 𝜖 occurs with powers ≥ k). The last equation obviously gives the TDSE for 𝜖 → 0.
- 39.
One sometimes reads the term ‘dislocalization’ in this connection, but it does usually more harm than good. In solid state physics, ‘dislocalization’ has the more specialized meaning of electron wave functions in partially filled bands of the solid having strongly overlapping supports, whence none is really ‘localized’ at a particular nucleus. This phenomenon is in fact connected to such familiar properties as conductivity (cf. Gross and Marx 2012, p. 134 ff.). This is to be sharply distinguished from ‘quantum nonlocality’ though, as we shall see in Chap. 4 (cf. also Zeh 2012, p. 84 on this potential confusion).
- 40.
Subtleties arise e.g. for Bloch waves in crystals (cf. Gross and Marx 2012, p. 341).
- 41.
For the given choices
or, with 
(cf. Basdevant and Dalibard 2002, p. 122). - 42.
It is obvious that Born had a deterministic notion of causality in mind, which we know is not necessarily always apt (e.g. Paul and Hall 2013, p. 63 ff., for some discussion).
- 43.
German: “Eine gewisse härte liegt ohne Zweifel zurzeit noch in der Verwendung einer komplexen Wellenfunktion.” (emphasis in original)
- 44.
Since the generalization to 3D is straightforward, we are here limiting our attention to one dimension, and we also omit the reference to the domain of integration for simplicity. We shall avail ourselves of both these simplifications more often in what follows.
- 45.
Schrödinger was, in fact, not fond of the idea of collapses at all, but sought for a theory purely in terms of waves. Besides the conflict with (almost) point-like measurements, he also wrestled with this difficulty of the spreading wave packet. In doing so, he discovered an outstanding example, the coherent states of the harmonic oscillator potential, which most closely mimic classical behavior. But the generalization was not straightforwardly possible, and the example remained a solitary one (cf. Bitbol 1996, p. 46; Schlosshauer 2007, p. 117).
- 46.
‘Inseparability’ can have multiple levels here: for one it can mean that a separation-ansatz for the TDSE does not work, in the sense that one cannot factor a solution of the TDSE for the different coordinates of one system. But the meaning of interest here concerns multiple systems and will be made precise only in Sect. 2.1.5.
- 47.
“[D]ie Schrödingerschen Wellen laufen ja gar nicht im gewöhnlichen Raume, sondern im ‘Konfigurationsraume’, der soviele Dimensionen hat, als die Anzahl der Freiheitsgrade des betrachteten Systems beträgt (3N-Dimensionen für N Partikel).” (Born 1926, p. 240)
- 48.
- 49.
More generally, the wave function of any single particle of spin s (\(=0,\frac {1}{2}, 1, \frac {3}{2}, \ldots \)) will be an element of the space \(L^{2}(\mathbb {R}^{3})\otimes \mathbb {C}^{2s+1}\), or equally \(L^{2}(\mathbb {R}^{3};\mathbb {C}^{2s+1})\). And for N indistinguishable particles, this will be \(L^{2}(\mathbb {R}^{3N};\mathbb {C}^{(2s+1)N})\) (cf. Gustafson and Sigal 2011, pp. 22 and 35).
- 50.
The complex of problems arising from this is incidentally one of the most thoroughly discussed ones in the philosophy of QM (traditionally in the context of Leibniz’ principle of the identity of indiscernibles). A nice historical and systematic overview can be found in Muller and Saunders (2008, pp. 505–508). See also Hawley (2009), and French’s and Krause’s (2006) comprehensive exposition of the topic for more details on the philosophical debate.
- 51.
For separable, countably infinite-dimensional spaces, the same can be established in terms of Fock space representations (cf. Horodecki et al. 2009, p. 918).
- 52.
We here appeal to Muller’s (2014, p. 426) version which is less ambiguous than the standard (textbook) one. The textbook version has it that no two fermions can be in the same state. But it has been emphasized (e.g. Muller and Saunders 2008, p. 511; Muller 2014, p. 422) that each fermion in a compound of multiple fermions of the same type may too occupy the ‘same state’ as given by its reduced density operator (see later).
- 53.
A potential counterexample is Cramer’s (1986) transactional interpretation which does interpret QM in terms of waves in spacetime—but on the cost of also accepting waves that can travel backwards in time. Most importantly, Cramer’s interpretation is riddled with difficulties, whence we deliberately choose not to bother with it any further here (for details, the reader is referred to Maudlin 2011, p. 180 ff.).
- 54.
To each of these supposedly intrinsic properties one could obviously object that their meaning is only defined w.r.t. the interaction of the system with some other system, which one could flesh out to yield a thorough relationalism.
- 55.
Note that Wallace’s (2016, p. 19) cosmological considerations hardly impair this point; Fuchs and Peres (2000) liken the required selective applications of QM in cosmology to “a few collective degrees of freedom” to applications of QM in SQUIDs (cf. later) and see “no difference in principle”. All evidence gathered about the universe requires ‘definite outcomes’ in some basis, so the orthodox interpretation or something closely related seems to be implicitly at play in the evaluation of the data.
- 56.
- 57.
Here we write SM to denote the otherwise unspecified composition of systems S and M; although cf. Greaves and Wallace (2013) for some insights on how systems compose.
- 58.
We stress that not every interaction can be described by a Hamiltonian of this kind, which has factorizing eigenstates. For a discussion see Joos et al. (2003, p. 48).
- 59.
In the non-ideal case where the system is demolished by the measurement, the evolution may usually be assumed to proceed in a similar fashion as
, with \(|\tilde {\alpha }_{k}|{ }^{2}\approx |\alpha _{k}|{ }^{2}\) and \({\vert \tilde {A}_{k}\rangle}\approx {\vert A_{k}\rangle}\) so that all that differs is a minor change in the coefficients and the state of the system (e.g. Bub 1997, p. 150). In the most drastic case though, the system gets destroyed (i.e., decomposed and/or absorbed into the apparatus). One then merely requires the apparatus states to carry, after the interaction, information about the state of S before its destruction (e.g. Wallace 2003, p. 420). - 60.
It is also a direct consequence of time evolution in the Heisenberg picture introduced later. A generalization to cases with complex environment-interactions is possible in terms of so called master equations (cf. Schlosshauer 2007, p. 154 ff.).
- 61.
The omitted computation is somewhat similar to that subsequent to equation (2.38).
- 62.
In case of a degenerate spectrum \(\sigma (\hat {O})\), one can sum over the n projectors onto eigenvectors spanning the subspace with eigenvalue o i, and then use a projection operator
hence defined instead (cf. Redhead 1987, p. 15). It is, however, crucial that the projectors 
are orthogonal in the sense that 
, in order for \(\hat {P}_{\lbrace o_{i_{j}}\rbrace }\) to be a projector itself. This also goes for a countably infinite subspace (‘n = ∞’) corresponding to the same eigenvalue (cf. Heinosaari and Ziman 2012, p. 23). - 63.
The details of mathematically generalizing the projection postulate to continuous spaces are quite intricate though (cf. Srinivas 1980), and we shall here restrict our attention to separable or finite dimensional spaces.
- 64.
For a few technical details see Appendix A.
- 65.
The notation may differ though, as can be seen from the discussion of the spectral decomposition in Appendix A. The example in Peres (2002, p. 386) is instructive w.r.t. the use of the above notation.
- 66.
For technical reasons, the two operators \(\mathbb {O}\) and
are also included in the set of properties as ‘trivial cases’ (cf. Busch et al. 1996, p. 10). - 67.
- 68.
- 69.
This does not preclude, however, the possibility of particles which always travel at superluminal velocities, so called tacyhons, which could not be decelerated to subluminal velocities instead (e.g. Maudlin 2011, p. 65 ff. for discussion). So far, however, there is no evidence for the existence of these; and there is even a sense in which they are incompatible with QFT qua localizable entities (cf. Sexl and Urbantke 1992, p. 27).
- 70.
Cf. in particular Maudlin (2011, p. 93 ff.) for an extensive discussion of different kinds of possible signals, and p. 2 ff. therein for a statement of the more general potential controversy.
- 71.
Whence the Lorentz transformations? A nice heuristic introduction can be found in Walker et al. (2012, chapter 37).
- 72.
Strictly speaking, η should be called a pseudo-metric tensor, since η(u, v) may yield values < 0. It is otherwise symmetric and linear in both arguments, and if \(\eta (u,v)=0, \forall u\in \mathbb {R}^{4}\), then v is the null vector (e.g. Nakahara 2003, p. 244).
- 73.
More precisely, a differentiable manifold is a topological space that is locally homeomorphic to \(\mathbb {R}^{4}\). A topological space is a set X, together with a collection of subsets of X which contains \(X, \varnothing \) and is closed under infinite unions and finite intersections. A homeomorphism is a continuous invertible map between topological spaces whose inverse is also continuous. A differentiable manifold M now is a topological space, endowed with a collection
(called an ‘atlas’) of pairs (called ‘charts’) of open sets U
i which jointly cover M (⋃iU
i = M) and homeomorphisms φ
i that map the U
i into open sets \(V_{i}\subseteq \mathbb {R}^{n}\), and for which, if \(U_{i}\cap U_{j}\neq \varnothing \) (i ≠ j), \(\varphi _{i}\circ \varphi _{j}^{-1}: \varphi _{j}(U_{i}\cap U_{j})\rightarrow \varphi _{i}(U_{i}\cap U_{j})\) is smooth. The φ
j equip the manifold locally (in the sets U
j) with coordinates (cf. Nakahara 2003, pp. 81, 85, and 171–172). - 74.
Note that we will sometimes use ‘QFT’ to mean the theoretical field in general, but also occasionally talk of ‘a QFT’, thereby meaning a concrete (heuristically) quantized field theory, such as e.g. QED.
- 75.
However, things are somewhat subtle here. Ruetsche presupposes the semantic view of theories (cf. Sect. 1.2), as did van Fraassen (1991) before her. On this account it is possible—at least in principle—to make two unitarily inequivalent theories come out physically equivalent, by assigning to them appropriately gerrymandered interpretations. Thus, physical equivalence becomes a question about fully interpreted physical theories (cf. Ruetsche 2011, p. 29).
- 76.
Note that without the hat on \(\hat {a}^{\dagger }_{\boldsymbol {p}_{i}}\) and † replaced by ∗ this would just be a (countable) Fourier expansion of a kind of quantum wave packet ψ ∗. In virtue of this, the operator \(\hat {\psi }^{\dagger }(\boldsymbol {x})\) can be viewed as a quantization of an object that is already the solution to a quantum mechanical equation (the ψ-function), which provides at least an intuitive (though supposedly historically inaccurate) explanation for the name ‘second quantization’.
- 77.
The factor \(\frac {1}{\sqrt {2E_{\boldsymbol {p}}}}\) is included to ensure Lorentz-invariance (cf. Lancaster and Blundell 2014, p. 101 ff. for discussion).
- 78.
In principle the meaning of the intrinsic handednesses is itself worth philosophizing about, since it may be taken to have implications, say, for substantivalism or relationalism about space (e.g. Earman 1991; Lyre 2005). But we will here rather concern ourselves with other matters more relevant for the discussion to come.
- 79.
For a very gentle start cf. Susskind and Hrabovsky (2013).
- 80.
Indeed, in even closer analogy to non-relativistic QM, \(\hat {\phi }(\boldsymbol {x})\) may just be represented as ‘multiplication by a scalar field ϕ(x)’, when operated on a quantum state |ϕ〉, and \(\hat {\pi }(\boldsymbol {x})\) as a functional derivative operator − iħ δ δϕ(x) w.r.t. ϕ(x). This is the functional Schrödinger representation (e.g. Hatfield 1992, p. 199 ff.), where the theory is taken to treat of states | Ψ〈 that have functionals 〈ϕ| Ψ(t)〉 = Ψ[ϕ, t] of field configurations ϕ as expansion coefficients (the ‘functional analogue’ of wave functions).
- 81.
For the notion of global hyperbolicity e.g. Smeenk and Wüthrich (2010, p. 593 ff.) or Ruetsche (2011, p. 107). Roughly, the idea is that there is a spacelike hypersurface Σ which has a future and past domain of dependence (D( Σ)+, D( Σ)−) such that the union of these is the whole spacetime \(\mathbb {M}\). The domains of dependence D( Σ)± are defined as sets of points p such that any past (+ ) or future (−) directed inextendible timelike curve through p has to intersect Σ, where timelike past/future directedness means that at any point the tangent vector of the curve falls into or on the past/future light cone, and inextendibility means that the curve has no endpoints.
- 82.
The metaphor “machine” for mathematical objects which take something in and give back an output can be traced back to John Wheeler’s use of the word for operators in QM (cf. Susskind and Friedman 2014, pp. 52–53).
- 83.
- 84.
Halvorson and Clifton (2002a, p. 207; emphasis in original) conclude the discussion of their results with the remark that relativistic QFT “does permit talk about particles—albeit, if we understand this talk as really being about the properties of, and interactions among, quantized fields.” But they concede that QFT only gains a capability of “explaining the appearance of macroscopically well-localized objects” in virtue of talk about the interactions of ‘quantized fields’ “modulo the standard quantum measurement problem[…]” (ibid.; my emphasis—FB), whose persistence in QFT is, of course, intimately connected to the mathematical representation of ‘quantum fields’ as operators. Our phenomenological particle˜ concept is untouched by all this, as will be explained in more detail later.
- 85.
Since \(\hat {a}_{\boldsymbol {p}}\hat {a}^{\dagger }_{\boldsymbol {p}} = [\hat {a}_{\boldsymbol {p}}, \hat {a}^{\dagger }_{\boldsymbol {p}}] + \hat {a}_{\boldsymbol {p}}^{\dagger }\hat {a}_{\boldsymbol {p}} = \delta _{\boldsymbol {p}\boldsymbol {p}} + \hat {a}_{\boldsymbol {p}}^{\dagger }\hat {a}_{\boldsymbol {p}}\), this immediately gives the desired expression.
- 86.
Teller (1995, p. 131), in particular, highlights that these non-vanishing vacuum expectation values (or ‘vacuum fluctuations’) are a consequence of the theory with normal ordering in place.
- 87.
- 88.
\(\mathbb {R}\) for ‘Rindler’, since the spacetime region in which the accelerated trajectory is located for the stationary observer is also called a Rindler wegde (cf. Crispino et al. 2008, p. 792 ff.).
- 89.
- 90.
We here emphasize again that the integrals are definite and x, y, z represent four-vectors, not Cartesian coordinates. The square of an integral means multiplying the integral by itself; but the integration variables can always be renamed individually (in each integral), whence in the square one will meet with an integration over two sets of variables. You can also think of it like this: A simple integral computes the area under a curve, so multiplying two integrals means computing a volume under two curves along different dimensions.
- 91.
Strictly speaking, this ‘renormalization group’ of scaling transformations is only a semigroup because there is not an inverse to every transformation (cf. Kadanoff 2013, p. 167). One also encounters talk of the ‘Wilsonian revolution’ (e.g Kadanoff 2013, p. 162), due to the importance of Kenneth Wilsons’s contributions to the field (e.g. Wilson 1971a,b).
- 92.
Λ is a momentum, so ħ∕ Λ defines a length scale that becomes smaller as Λ. Λ →∞ means ‘increasing the fineness of grain’ of the QFT.
- 93.
More precisely, it is a consequence of Haag’s theorem that if one holds that there be unique vacuum states | Ω0〈, | Ωλ〈 for free and interacting theories respectively, and if one requires that these be invariant under unitaries such as \(\hat {U}(\boldsymbol {a})\) representing Eucildean symmetries such as a translation by a, then given that one also allows initial and final scattering states to be asymptotically free, an interacting theory such as the \(\hat {\phi }^{4}\)-theory discussed above can be demonstrated to be unitarily equivalent to the free theory, which one can take to mean that it is no interacting theory after all, i.e. that (given the previous assumptions) there are no interacting theories (cf. Haag 1996, p. 55; cf. also Ruetsche 2011, pp. 251–253 and Teller 1995, pp. 115–116 and 122–123 for further discussion).
- 94.
Talk of a ‘physical world’ may suggest that there is a ‘non-physical world’ as well, maybe a ‘mental’ one. But we do not intend to take sides on this issue here at all. Purely on the level of scientific description, it is not the case that physical and mental phenomena do coincide. That is, to date there is no fully physical theory of the mind, and if ‘the hard problem’ (Chalmers 1996, p. xii) of why there even is an experienced inner life accompanying neurophysical processes is indeed as hard as it seems, it is not clear that there ever will be. But again, we are here only suggesting a prima facie non-identity of physical and mental (or social, or economical, or…) phenomena, not serious mind-body dualism.
, which, however, in the present case of a single wave coincides with the phase velocity, so that no harm comes from the simplification. Note that no energy or information can be transported with phase velocity: pictorially it describes how fast the ‘ripples’ in the packet propagate, but the ripples ‘diminish’ while approaching the boundary of the packet, and so no energy or information is transmitted with a speed > c (e.g. Griffiths 
, one has 
which would equally result from 
with 
the density operator of a proper mixture (cf. 
replaced by 
, and where 𝜖 is the so called Levi-Civita symbol which gives back 1 if k, ℓ, and m are cyclical permutations of 1, 2, 3, − 1 for anti-cyclical ones, and 0 if two of the numbers are identical. In virtue of this property, the Pauli matrices form an abstract commutator algebra (a Lie algebra), as do other types of angular momentum in QM (e.g. Schwindt 
there are simultaneous eigenvectors 
on a space of functions of position, ψ(x), and 
. Note that in a space of momentum-dependent functions, 
or, with 
(cf. Basdevant and Dalibard 
, with 
hence defined instead (cf. Redhead 
are orthogonal in the sense that 
, in order for 
are also included in the set of properties as ‘trivial cases’ (cf. Busch et al. 
(called an ‘atlas’) of pairs (called ‘charts’) of open sets U
i which jointly cover M (⋃iU
i = M) and homeomorphisms φ
i that map the U
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Boge, F.J. (2018). Some Quantum Mechanics, Its Problems, and How Not to Think About Them. In: Quantum Mechanics Between Ontology and Epistemology. European Studies in Philosophy of Science, vol 10. Springer, Cham. https://doi.org/10.1007/978-3-319-95765-4_2
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