Time-energy uncertainty does not create particles
aa r X i v : . [ phy s i c s . h i s t - ph ] J u l Time-energy uncertainty does not create particles
B W Roberts and J Butterfield Philosophy, Logic & Scientific Method, Centre for Philosophy of Natural and Social Sciences,London School of Economics & Political Science, London, WC2A 2AE, UK Trinity College, University of Cambridge, Cambridge, CB2 1TQ, UKE-mail: [email protected], [email protected]
Abstract.
In this contribution in honour of Paul Busch, we criticise the claims of many expo-sitions that the time-energy uncertainty principle allows both a violation of energy conservation,and particle creation, provided that this happens for a sufficiently short time. But we agree thatthere are grains of truth in these claims: which we make precise and justify using perturbationtheory.8 July 2020. Forthcoming in
Journal of Physics: Conference Proceedings , MathematicalFoundations of Quantum Mechanics in Memoriam Paul Busch
1. Introduction
In expositions of quantum theory, it is often said that energy conservation can be violated, andthat particles can ‘pop in to existence’ out of nowhere, thanks to a time-energy uncertaintyprinciple. Time-energy uncertainty was, of course, one of Paul Buschs areas of expertise. Hisinsightful analyses of it began already in his two 1990 works [4, 5] and the topic remained anabiding interest of his, shown for example in his 2008 review article about time in quantumphysics [6]. It seems to us very likely that Busch, with his clarity and precision of thought,would have had misgivings about this “folklore”. Accordingly, we propose to commemorate hislife and work by criticising it—in a way that we hope is worthy of his memory. But we will alsoargue that the folklore contains grains of truth: which we will make precise and justify, usingerturbation theory.We begin with some illustrative quotations. Thus Jones proposes that,“a consequence of the Heisenberg Uncertainty Principle is that we can take seriouslythe possibility of the existence of energy non-conserving processesprovided the amountby which energy is not conserved, E violation , exists for a time less than t = ~ / E violation ”[12, p.226]This folklore is not just a myth of pedagogy or popular exposition. Many excellent textbooks make similar comments; such as the classic quantum field theory textbook of Peskin andSchroeder:“Even when there is not enough energy for pair creation, multiparticle states appear,for example, as intermediate states in second-order perturbation theory. We can thinkof such states as existing only for a very short time, according to the uncertaintyprinciple ∆ E · ∆ t = ~ . As we go to higher orders in perturbation theory, arbitrarilymany such ‘virtual’ particles can be created.” [18, p.13]Similarly, in their textbook on general relativity, Hobson et al. [11] say that time-energyuncertainty is responsible not just for virtual particles, but for concrete physical predictionslike Hawking radiation :“Hawking’s original calculation uses the techniques of quantum field theory, butwe can derive the main results very simply from elementary arguments. ... Paircreation violates the conservation of energy and so is classically forbidden. In quantummechanics, however, one form of Heisenberg’s uncertainty principle is ∆ t ∆ E = ~ ,where ∆ E is the minimum uncertainty in the energy of a particle that resides in aquantum mechanical state for a time ∆ t . Thus, provided the pair annihilates in a time As Fermat might say: we have discovered a marvellous number of comments of this kind, which this footnoteis too small to contain. Here are a couple from books about quantum field theory; though one of our main pointswill be that the issues are not specific to quantum field theory, but arise already in quantum mechanics. The firstbook is about quantum field theory’s philosophical interpretation, the second about its mathematics. Teller [19,p.148] remarks that violation of local energy conservation is “customarily excused” by the time-energy uncertaintyprinciple; while Folland [8, p.133] writes that, “the uncertainty principle allows the particle and/or quantum tobe temporarily ‘off mass-shell’ between the times of emission and absorption or vice versa”. There is a sense in which Hawking radiation [10], when viewed as a comparison between a quantum fieldtheory constructed at past null infinity and one constructed at future null infinity, is indeed associated withparticle creation. However, this is a matter of inequivalent vacua, and so spontaneous symmetry breaking, not oftime-energy uncertainty. t = ~ / ∆ E , where ∆ E is the amount of energy violation, no physical lawhas been broken.” [11, § E , as longas you ‘pay it back’ in a time ∆ t ∼ ~ / E ; the greater the violation, the briefer theperiod over which it can occur. There are many legitimate readings of the energy-timeuncertainty principle, but this is not one of them. Nowhere does quantum mechanicslicense violation of energy conservation”. [9, p.115]We agree with Griffiths. Of course, mass-energy is exactly conserved in an isolated physicalsystem, in quantum physics no less than classical physics. In quantum theory, the unitarypropagator U t (a strongly continuous unitary representation of the reals under addition) can bewritten U t = e − itH , where the self-adjoint generator H is the energy. And, for any initial state ρ that evolves unitarily according to ρ t = U t ρU ∗ t , the energy expectation value does not changeover time, since H and U t commute: Tr( ρ t H ) = Tr( ρU ∗ t HU t ) = Tr( ρH ) for all t ∈ R .Nevertheless, what the folklore says about time, energy, and energy conservation containssome grains of truth. We will focus on three ideas:(i) ( non-conservation ) There is some sense in which ‘energy’ associated with a perturbedsystem is not conserved;(ii) ( particle creation ) There is some sense in which that non-conservation allows the non-conservation of particle-number; and(iii) ( shorter times ) There is some sense in which more particle creation occurs during shortertimes.So our aim is to do the exercise of making these statements precise, and verifying them. Ourlesson will be that, in each case, it is not a time-energy uncertainty relation that provides thewiggle-room to create particles: hence our slogan, “time-energy uncertainty does not createparticles.” The particles are rather best viewed as artefacts of the shifted perspective one Another dissenter is Bunge [3], who gives a short but scathing criticism of understanding virtual particles interms of time-energy uncertainty.
2. The perturbation view of virtual states
For our purposes, a quantum system is a triple ( H , A , t U t ), where H is a separablecomplex Hilbert space, A is a von Neumann algebra of linear operators on H , and t U t is a strongly continuous one-parameter unitary representation in A of the group R underaddition. The Hilbert space represents a collection of states; the algebra contains a collectionof operators that represent observables; and the unitary representation provides the dynamics.By Stone’s theorem, there exists a unique self-adjoint operator H ∈ A (the ‘energy’ operator or‘Hamiltonian’) such that the dynamics can be written in the form, U t = e − itH , for all t ∈ R .Perturbation theory uses a quantum system that is in some way tractable to approximatea quantum system that is not. For example: the energy H of a Helium atom, which containstwo mutually repulsive electrons, can be approximated by a system whose energy H ignoresthe mutual repulsion of the electrons. In such examples, one begins with a quantum system( H , A , t U t ) and a set of operators { A λ } ⊂ A parametrised by a real number λ , one value ofwhich gives the correct, or physically real, operator of interest: A λ = A + V λ . (1)This set is constructed in such a way that as λ →
0, we get A λ → A in the operator norm. A is called the ‘unperturbed’ operator, and the set A λ is called a ‘perturbation’. The hope is boththat a physical system can be correctly described by A λ for some value of λ , and also that itsproperties can be accurately approximated using known facts about the more tractable operator A .Perturbation theory allows one to approximate various aspects of the operator A λ when it canbe represented in a power series expansion around λ = 0. That is, one seeks an exact expression4f A λ of the form, A λ = A + λ (cid:18) ddλ V λ (cid:19) (cid:12)(cid:12)(cid:12) λ =0 + λ (cid:18) d dλ V λ (cid:19) (cid:12)(cid:12)(cid:12) λ =0 + · · · (2)The n th-order approximation of A λ is by definition the sum of the first n terms in this series. As n → ∞ , the series approaches A λ in the operator norm. A wide class of problems can be solvedby adopting the simple approximation where only the first two terms are calculated. Defining V := ddλ V λ (cid:12)(cid:12)(cid:12) λ =0 this gives what is called a ‘linear’ perturbation: A λ ≈ A + λV. (3)The eigenvalue problem for an operator expressed as a linear perturbation can typically be givenan approximate analysis thanks to classic results in perturbation theory. A typical way for virtual states to arise in perturbation theory is in its application to thedynamics. Let ( H , A , t U t ) be a quantum system, and consider a second one-parameterunitary representation t U t . We write H and H for their respective Hamiltonian generators,and refer to the former as the ‘perturbed’ (or ‘interacting’) Hamiltonian, while the the latteris the ‘unperturbed’ Hamiltonian. We define V := H − H and refer to it as the ‘potential’.Writing U t = U t ( U − t U t ), we expand the term in parentheses as a power series around t = 0, U t = U t (cid:18) I + t ddt ( U − t U t ) (cid:12)(cid:12)(cid:12) t =0 + t d dt ( U − t U t ) (cid:12)(cid:12)(cid:12) t =0 + · · · (cid:19) . (4)To the extent that t is close to zero and V = H − H is small (in the operator norm), theoperator U t is approximated by summing the first N terms in this series and ignoring the rest[13, § II.3]. Note however that this partial sum of the first N terms is in general not unitary.Writing U ( n ) t to denote the n th term in the series, the first few orders of approximation can For example, if it is the case that for all λ , A λ = A + λV has a discrete spectrum and commutes with itsadjoint ( A λ A ∗ λ = A ∗ λ A λ ), then by the Mitzkin-Taussky theorem, the eigenvalues of a iλ of A λ are linear in λ (inthat a iλ = a i + λk i ), and its eigenvectors are analytic functions of λ [13, § λ = 0 to give an approximation in terms of (usually already-known) eigenvectors of A .
5e calculated by applying the Leibniz rule to the derivatives: U (0) t = U t U (1) t = − itU t VU (2) t = t U t (cid:0) [ V, H ] − V (cid:1) (5)The virtual states picture arises from imagining that each contribution to the series isthe amplitude of a “scattering event” in its own right. For example, suppose ψ a and ψ b areorthogonal eigenvectors of the unperturbed Hamiltonian H , and that we wish to approximatethe amplitude h ψ b , U t ψ a i associated with a transition during time t from ψ a to ψ b . In a first-order approximation, we replace U t with the sum of the terms U (0) t + U (1) t . In a more accuratesecond-order approximation, we replace it with U (0) t + U (1) t + U (2) t , and so on. With a littlecalculation , the terms in these approximations are found to be, h ψ b , U (0) t ψ a i = 0 h ψ b , U (1) t ψ a i = − ite ibt h ψ b , V ψ a ih ψ b , U (2) t ψ b i = e ibt ( a − b ) h ψ b , V ψ a i − e ibt h ψ b , V ψ a i (6)where a and b are the H eigenvalues associated with ψ a and ψ b , respectively. The zeroth-ordercontribution h ψ b , U (0) ψ a i , viewed as a transition amplitude in its own right, describes an initialstate ψ a that never transitions to ψ b . The first-order contribution might lead us to say thatthe presence of the potential V ‘deflects’ the initial state ψ a to ψ b with some probability. Thesecond-order contribution has two terms, the first of which will just get collected together withthe first-order one in the series, and the second of which is a ‘deflection’ by the potential V . Ofcourse, we have not given any reason to view this language as anything more than short-hand;strictly speaking, each is simply a contribution to a series that approximates h ψ b , U t ψ a i .Virtual states arise as ‘intermediate states’ in this kind of analysis. In the example just givenof approximating the transition amplitude h ψ b , U t ψ a i , they begin to appear at the level of thesecond-order contribution U (2) t . Instead of thinking of the second-order term e ibt h ψ b , V ψ a i as a Zeroth order: h ψ b , U (0) t ψ a i = e iat h ψ b , ψ a i = 0. First-order: h ψ b , U (1) t ψ a i = − it h ψ b , U t V ψ a i = − ite ibt h ψ b , V ψ a i .Second-order: h ψ b , U (2) ψ b i = h ψ b , U t [ V, H ] ψ a i − h ψ b , U t V ψ a i = e ibt ( a − b ) h ψ b , V ψ a i + e ibt h ψ b , V ψ a i . V , let us rewrite it in the form, e ibt h ψ b , V ψ a i = h ψ b , V e ib ( t − t ′ ) ψ ′ ih ψ ′ , V e ibt ′ ψ a i (7)where we define ψ ′ := | V ψ a | V ψ a , and choose any t ′ such that 0 < t ′ < t . Now, instead ofviewing the transition as going from ψ a to ψ b in the potential V during a time t , we can viewit as consisting of an intermediate transition from ψ a to ψ ′ in V during time t ′ , followed by atransition from ψ ′ to ψ b in V during the later time-interval t − t ′ . The intermediate state ψ ′ is an example of a virtual state . As expected, the third-order transitions have a V term thatgives rise to a pair of virtual states, and so on up the series.One can draw a Feynman diagram for each contribution h ψ b , U ( n ) t ψ a i in this series, illustratedfor U (0) , U (1) , and U (2) in Figure 1. Virtual states appear, beginning in the second-orderdiagrams, as states without open endpoints, such as ψ ′ in the right-most diagram of the Figure.This encodes the fact that they are not associated with the measured in-state or out-state inthe scattering experiment associated with this amplitude. However, again: we have given noreason to view each diagram as anything other than shorthand for a term h ψ b , U ( n ) ψ a i in a seriesapproximation of the amplitude h ψ a , U t ψ b i ≈ h ψ a , U (0) t ψ b i + h ψ a , U (1) t ψ b i + h ψ a , U (2) t ψ b i + · · · . ψ a V ψ a ψ b VV ψ b ψ a ψ ′ Figure 1.
Feynman diagrams for the first three terms in a perturbation series for h ψ b , U t ψ a i , with U t = e − it ( H + V ) and ψ a , ψ b eigenvectors of H . The second-order state ψ ′ is ‘virtual’.
3. The appearance of energy non-conservation
To sum up: the perturbation view is one of shifting perspectives. We describe a quantum system( H , A , t U t ) from the perspective of the n th-order approximation in a perturbation series,recognising that on this perspective, the system will sometimes appear to deviate from its true7perturbed’ dynamics, as well as from the idealised ‘unperturbed’ dynamics.As it happens, one such deviation is that energy conservation can appear to fail. Thus wehave our first claim:(i) ( non-conservation ) There is some sense in which ‘energy’ associated with a perturbedsystem is not conserved.In the context of the virtual states described in the previous section, the ‘energy’ that fails to beconserved is associated with the idealised Hamiltonian H of the unperturbed dynamics. In thezeroth-order approximation U (0) , it is conserved. However, in the first-order and second-orderapproximations, it is not, in that there are eigenstates of H that are not stationary. This is dueto the fact that for interacting systems with [ H , V ] = 0, the unperturbed Hamiltonian is notstationary, U t H U ∗ t = 0, or equivalently, [ H, H ] = 0. So this is one thing that could be meantby saying ‘energy’ is not conserved in the presence of virtual states.However, in a quantum system ( H , A , t U t ), it is the generator H of the true i.e. perturbeddynamics U t that represents the ‘true energy’ of the system, not the idealised Hamiltonian H . So, a more physically interesting question is whether H commutes with the approximatedynamics given by a partial sum, up to say the N th term, of the perturbation series, eq. 4 and5. Of course, it may happen that for some finite N , we find that P Nn U ( n ) = U t , so that P Nn U ( n ) is an exact description rather than an approximate one, and the energy H is indeed conserved.But in general, the approximate dynamics P Nn U ( n ) is not equal to U t , indeed is not even unitary;and H will not be conserved under it.On the other hand, there is a sense in which energy H for the perturbed dynamics U t comescloser to being conserved by the approximate dynamics, the higher the order of approximation.For as we add more terms, the resulting approximate dynamics P n U ( n ) t better approximatesthe true dynamics U t ; that is, P Nn =0 U ( n ) t → U t in the operator norm as N becomes arbitrarilylarge. But this implies: hP Nn =0 U ( n ) t , H i → N → + ∞ , (8)giving perfect conservation of energy in the limit. This is ironic, in that Peskin and Schroederwrite: “As we go to higher orders in perturbation theory, arbitrarily many such ‘virtual’ particles In general, k A n − B k → k [ A n , B ] k →
0, since k [ A n , B ] k = k ( A n − B ) B + B ( B − A n ) k ≤k A n − B kk B k + k B kk B − A n k = 2 k B kk A n − B k .
4. Particle creation
We now turn to the statement of (ii), the particle-number claim. For this we need to addsome notion of particle number to our description. This will consist in a representation ofannihilation ( a i ) and creation ( a ∗ i ) operators on H for i ∈ Z + , which satisfy [ a i , a ∗ j ] = δ ij and[ a i , a j ] = [ a ∗ i , a ∗ j ] = 0. Let N = P i a ∗ i a i be the ‘particle number’ operator. Taking U t as theunperturbed dynamics associated with no interactions, we assume for simplicity that [ N, U t ] = 0,and hence that the unperturbed system is one in which this (unperturbed) particle-number isconserved. However, if [ N, U t ] = 0, then particle number will in general not be conserved bythe ‘true’ dynamics; and in general, it will not be conserved under the dynamics given by anyof the partial-sum approximations P n U ( n ) t . (Agreed, N might be conserved by U t and-or by anapproximate dynamics. But this will not hold true in general.)That is, we have:(ii) ( particle-creation ) The dynamics generated by the partial-sum approximation P n U ( n ) t doesnot in general conserve the unperturbed particle number N .
5. Shorter times
We now turn to our final claim:(iii) ( shorter times ) There is some sense in which more particle creation occurs during shortertimes.Here we enter the realm of the time-energy uncertainty principle—or, rather, principles—that areinvoked by the cavalier textbook tradition with which we began. These principles are surveyedby Busch [4, 5, 6]. For us, there are two main points to make, corresponding to two broadunderstandings of time-energy uncertainty. The first point is not connected to perturbationtheory: it is really a warning against an untenable understanding of time-energy uncertainty.The second point will be more positive, in that it will vindicate the shorter-time claim (iii).9he first point concerns what Busch [6] suggests one should call external time (or in [4]:‘pragmatic time’): namely, time as measured by clocks that are not coupled to the objectsstudied in the experiment. So in this role, time specifies a parameter or parameters of theexperiment: e.g. an instant or duration of preparation or of measurement, or the time-intervalbetween preparation and measurement. In this role, there seems to be no scope for uncertaintyabout time. And indeed, our first point here is a warning—following Busch [4]. For as Buschdiscusses, there is a tradition (deriving from the founding fathers of quantum theory) of anuncertainty principle between:(1) the duration of an energy measurement; and(2) either the range of an uncontrollable change of the measured systems energy, or theresolution of the energy measurement, or the statistical spread of the systems energy.To give a little more detail: Busch [4, § § P ) An energy measurement of duration ∆ t leads to an uncontrollable and unpredictablechange of the (previously sharply defined) energy by an amount of the order ∆ E such that∆ E. ∆ t ≥ ~ ; so that there is no short-time reproducible (first kind) energy measurement.( P ′ ) An energy measurement of duration ∆ t must carry an inaccuracy ∆ E such that theuncertainty relation ∆ E. ∆ t ≥ ~ is satisfied.Busch argues, and we agree, that Aharonov and Bohm [1] refute this tradition; (see [4], especially §
4, and [6, § This leaves little room for ‘energyuncertainty’ to develop during brief (in external time) energy measurements, either in terms ofuncontrollable changes in energy, or in terms of measurement inaccuracy.Our second, more positive, point concerns what Busch [6] suggests one should call ‘intrinsictime’ (or in his [4]: ‘dynamical time’): namely, a dynamical variable of the target system itself Note that Busch argues that a proper analysis and vindication of Aharonov and Bohms refutation usespositive operator-valued measures (POVMs) to describe measurement outcomes, a notion of physical quantitythat generalises projection-valued measures (PVMs); this follows the tradition of Ludwig [16, 17], and is developedat book-length by Busch et al. [7]. A ∈ A and density operatorstate ρ defines a characteristic time interval, τ ρ ( A ), in which the expectation value changes‘significantly’. For example: in the Schr¨odinger representation on the space of L ( R ) functions,if Aψ ( x ) := Qψ ( x ) = xψ ( x ) for all ψ ( x ) in the domain of Q and if ρ = E ψ is the projectionassociated with a wave packet ψ ( x ), then τ ρ ( A ) could be defined as the time interval requiredfor the bulk of the wave packet to shift by its width—in some sense of ‘width’.Various definitions of such a time interval are available. We will choose what is sometimescalled the ‘characteristic time’ associated with the dispersion of an operator-state pair. Thisobeys what is probably the best-known time-energy uncertainty principle for intrinsic times: the Mandelstam-Tamm uncertainty principle .One arrives at it for a quantum system ( H , A , t U t ) by combining three ideas: (a) theHeisenberg equation of motion for an operator A ∈ A with A ( t ) := U t AU ∗ t , i ~ ddt A ( t ) = [ H, A ( t )]; (9)(b) the Heisenberg-Robertson uncertainty principle , that for any quantities A, B ∈ A , anddensity operator (quantum state) ρ ,∆ ρ A ∆ ρ B ≥ | Tr( i [ A, B ] ρ ) | ; (10)and (c) the definition of a characteristic time for an operator A that does not commute with theunitary dynamics t U t , and for a time-independent state ρ that is not an eigenstate of A : τ ρ ( A ) := ∆ ρ A | ddt Tr( A ( t ) ρ ) | , (11)i.e. the time it takes for the expectation value of A to change by its standard deviation. Fromthese definitions it immediately follows that, τ ρ ( A )∆ ρ ( H ) ≥ ~ , (12) Cf. [2, Theorem 8.1.2]
A, H ] = 0 and ρ is not an eigenstate of H .This principle can be applied in a perturbation theory analysis of a quantum system( H , A , t U t ) with Hamiltonian generator H and an unperturbed Hamiltonian H , in twoways, so as to give two construals of the shorter times claims (iii). The first way puts H for A in eq. 12, while the second way puts a number operator N for A .So first set A := H in eq. 12, to get: τ ρ ( H )∆ ρ ( H ) ≥ ~ . Suppose that ∆ ρ H is very small,corresponding to a state ρ that is ‘peaked’ in unperturbed energy, so that the characteristic time τ ρ ( H ) is comparatively small. The Mandelstam-Tamm uncertainty principle then implies that∆ ρ H is comparatively large, and hence that the spread of the system’s true (unperturbed) energyis large. In this sense, a short characteristic time and a peaked distribution of the unperturbedenergy is associated with a large uncertainty of the system’s true energy. Thus we have onegeneral way to construe the shorter-times claim (iii) above. Agreed, this is a construal that usesthe unperturbed Hamiltonian H , not an (unperturbed) number operator N .And finally: a related construal of claim (iii) can be given, about a particle number opera-tor N associated with the unperturbed dynamics i.e. that satisfies [ N, H ] = 0; so that underunperturbed dynamics, this particle number is conserved. Such an operator will typically notbe conserved by the true dynamics, in that [ H, N ] = 0. Thus let us suppose now that ρ is astate for which the true i.e. perturbed energy H is peaked, in that ∆ ρ H is small, and hence(eq. 12) that the characteristic time (under the true i.e. perturbed dynamics) τ ρ ( N ) is large.Then, the larger | ddt Tr( N ( t ) ρ ) | is, corresponding to a fast rate of change of the expectationvalue of N in the true dynamics, the larger the spread ∆ p N must be, in order to guarantee that τ ρ ( N ) := ∆ ρ N/ | ddt Tr( N ( t ) ρ ) | (cf. eq. 11) is sufficiently large. So this construal of claim (iii)amounts to: ‘If the perturbed energy H is peaked, and the expectation value of N changes fast(“short times”), then N has large spread (“non-negligible amplitudes for values far from theexpectation value”)’.To conclude: we hope to have shown that with a little Buschian wisdom , one can recover somegrains of truth from some cavalier statements in the textbook tradition. May Paul’s legacycontinue to inspire our community to emulate his craftsmanship and creativity.12 cknowledgments
We thank Klaas Landsman, Jos Uffink and Reinhard Werner for advice and encouragement atan early stage; two anonymous referees for helpful comments and corrections; and the editors,not least for their patience. Bryan W. Roberts was supported by the Leverhulme Trust’s PhilipLeverhulme Prize and as a Visiting Fellow Commoner at Trinity College, Cambridge.
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