Quantum mechanics of 4-derivative theories
aa r X i v : . [ h e p - t h ] A p r Quantum mechanics of 4-derivative theories
Alberto Salvio a and Alessandro Strumia b a Departamento de F´ısica Te´orica, Universidad Aut´onoma de Madridand Instituto de F´ısica Te´orica IFT-UAM/CSIC, Madrid, Spain b Dipartimento di Fisica dell’Universit`a di Pisa and INFN, ItaliaCERN, Theory Division, Geneva, Switzerland
Abstract
A renormalizable theory of gravity is obtained if the dimension-less 4-derivativekinetic term of the graviton, which classically suffers from negative unboundedenergy, admits a sensible quantization. We find that a 4-derivative degree of free-dom involves a canonical coordinate with unusual time-inversion parity, and thata correspondingly unusual representation must be employed for the relative quan-tum operator. The resulting theory has positive energy eigenvalues, normalizablewave functions, unitary evolution in a negative-norm configuration space. Wepresent a formalism for quantum mechanics with a generic norm.
Contents
Introduction
Newton invented classical mechanics putting two time derivatives in his equation F = m ¨ x ,which corresponds to a kinetic energy with 2 time derivatives, m ˙ x / . Later Ostrograd-ski proofed a no-go theorem: non-degenerate classical systems with more than two timederivatives contain arbitrarily negative energies and develop fatal run-away instabilities [1].Classically, they do not make sense.The discovery that nature is relativistic and quantum opened the quest for an extension ofNewtonian gravity. A century ago Einstein and Hilbert found the classical theory of relativisticgravity. However, its quantum version is not renormalizable in 3+1 space-time dimensions.Sticking to the observed number of space-time dimensions, a renormalizable extension ofgeneral relativity is found by adding terms quadratic in the curvature tensor to the Einstein-Hilbert Lagrangian, such that the graviton acquires a 4-derivative kinetic term. Stelle pro-posed and dismissed this extension [2] (see also [3–9]).Recently the Higgs mass hierarchy problem brought interest to dimension-less theories.In this context, gravitons must have dimension 0 (being a dimension-less metric) and thereby must have a 4-derivative kinetic term. If these theories could make sense at quantum level,despite the negative classical energy, a great deal would be gained: relativistic quantumgravity, plus hierarchies among dynamically generated mass scales [8], plus inflation [10,11].Quantization can eliminate arbitrarily negative classical energies. The following exampleis well known: the classical relativistic spin 1/2 field is described by a spinor Ψ( x ) with DiracLagrangian L = ¯Ψ( i /∂ − m )Ψ containing one time derivative. Treating Ψ as a classical field(as initially proposed by Schr¨odinger), and inserting the plane-wave expansion Ψ( x ) = Z d p (2 π ) p E p [ a p,s u p,s e − ip · x + b † p,s v p,s e ip · x ] (1)in the Hamiltonian one finds negative energies in half of the configurations space: H = Z d p (2 π ) E p [ a † p,s a p,s − b p,s b † p,s ] , E p = p m + ~p (2)This classical arbitrarily negative energy is avoided by quantization with anti-commutators, ifthe vacuum state is appropriately chosen. Indeed, the two-state solution to { b, b † } = 1 showsthat one can switch annihilation with creation operators by choosing the vacuum to be thestate with lower energy.The spin 0 and spin 1 relativistic fields (described by dimension-1 fields with 2 deriva-tives) do not have this issue: the negative-frequency solutions to the Klein-Gordon equationcorrespond to Hamiltonians with positive energy.The general lesson is that quantization depends on the number of time derivatives. Unlike in the case of the Hydrogen atom emphasized by Woodard [12], where the instability eliminated byquantum mechanics occurs only in one point of the configuration space. T . In section 4 we recallthat a 4-derivative degree of freedom q ( t ) is described by two canonical coordinates: q = q and q = ˙ q . While q is T -even as usual, q is T -odd: we argue that thereby it naturally followsthe negative-norm representation. The resulting quantum theory is unitary: time evolutionpreserves the negative norm. The path-integral formulation is discussed in section 5. In sec-tion 6 we discuss the interacting theory, outline the extension to quantum field theory, anddiscuss the issue of giving a sensible interpretation to negative norms, via a postulate thatgeneralizes the Born rule. Conclusions are given in section 7. Let us now introduce the main issues in the simplest relevant case. Our final goal will be4-derivative gravity; however the graviton components can be Fourier expanded into modeswith given momentum and 4 time derivatives, and at leading order in the perturbative ex-pansion one has decoupled harmonic oscillators. So, we start considering a single mode q ( t ) ,described by the Lagrangian L = − ¨ q ω + ω ) ˙ q − ω ω q − V ( q ) = − q ( d dt + ω )( d dt + ω ) q − V ( q )+ total derivatives . where V ( q ) is some interaction. We assume real ω , ω , because we are interested in ghosts(negative kinetic and potential energy), not in tachyonic instabilities (potential unstable withrespect to the kinetic term). The − sign means that the ghost is the state with larger ω ; wechoose ω > ω and don’t explicitly discuss here the degenerate case ω = ω .Ostrogradski introduced an auxiliary coordinate q that allows to describe the 4-derivativeoscillator in canonical Hamiltonian form (see also ref. [16] for a review of this method): q = q, p = δ L δ ˙ q = ( ω + ω ) ˙ q + ... q ,q = λ ˙ q, p = δ L δ ˙ q = − ¨ qλ , (3)3here for a generic variable x we have introduced the variational derivative δ L δx = ∂ L ∂x − ddt ∂ L ∂ ˙ x + d dt ∂ L ∂ ¨ x + · · · . (4)While Ostrogradski assumed λ = 1 , we introduced an arbitrary constant λ . The system ineq. (3) can be solved for q and its time derivatives, q = q , ˙ q = q λ , ¨ q = − λp , ... q = p − (cid:0) ω + ω (cid:1) q λ , (5)and the Hamiltonian turns out to be H = X i =1 p i ˙ q i − L = p q λ − λ p − ω + ω λ q + ω ω q + V ( q ) . (6)In view of its first term, the classical Hamiltonian H has no minimal energy configuration:this is the essence of the Ostrogradski no-go classical theorem. Using the Poisson parentheses { , } one computes the Hamiltonian equations of motion: ˙ q = { q , H } = ∂H∂p = q λ , ˙ p = { p , H } = − ∂H∂q = − ω ω q − V ′ ( q ) , ˙ q = { q , H } = ∂H∂p = − λ p , ˙ p = { p , H } = − ∂H∂q = − p λ + ω + ω λ q . (7)For any λ they imply the classical Lagrangian equation of motion. Setting V = 0 , it is ( d dt + ω )( d dt + ω ) q = d qdt + ( ω + ω ) d qdt + ω ω q = 0 . (8)The corresponding classical solution, for given initial conditions q , ˙ q , ¨ q , ... q at t = 0 , is q ( t ) = − ω q + ¨ q ω − ω cos( ω t ) + ω q + ¨ q ω − ω cos( ω t ) − ω ˙ q + ... q ω ( ω − ω ) sin( ω t ) + ω ˙ q + ... q ω ( ω − ω ) sin( ω t ) . (9)This is a well behaved oscillator without run-away issues because the positive-energy andnegative-energy components are decoupled. Run-away solutions appear when they interactthrough a generic interaction, such as a V = 0 . The classical equation differs from the usual 2-derivative equation d ( q + ip ) /dt = iω ( q + ip ) ,so that, trying to quantize the theory, we do not define the usual annihilation operator a i ∝ q i + ip i . Rather, it is convenient to define the operators a i as the coefficients of given frequency: q ( t ) = a e − iω t + a e − iω t + h.c. (10)4he a , a can be expressed in terms of canonical Hamiltonian coordinates: a = λω ω q − iω q + ip λ − ω p λ λω ( ω − ω ) , (11) a = λω ω q − iω q + ip λ − ω p λ λω ( ω − ω ) . (12)Using the canonical quantization [ q i , p j ] = iδ ij one finds the commutation relations for the a i : [˜ a , ˜ a † ] = − , [˜ a , ˜ a † ] = 1 , all other commutators vanish. (13)having normalised ˜ a = p ω ( ω − ω ) a and ˜ a = p ω ( ω − ω ) a . The Hamiltonian is H = − ω ˜ a ˜ a † + ˜ a † ˜ a ω ˜ a ˜ a † + ˜ a † ˜ a . (14)The state 1 with higher frequency ω > ω is a ghost.As better discussed later in section 3.3, this system can be quantized in two different ways:1. Positive norm, negative energy . One redefines ˜ a ′ = ˜ a † , such that it has the usualcommutation [˜ a ′ , ˜ a ′† ] = 1 . The vacuum state | ˜0 i is defined as usual by ˜ a ′ | ˜0 i = 0 and ˜ a | ˜0 i = 0 . By solving this condition as a differential equation for ψ ˜0 ( q , q ) = h q , q | ˜0 i with p i = − i∂/∂q i one obtains the ground state wave function: ψ ˜0 ( q , q ) = exp (cid:18) − q ω ω + q /λ ω − ω ) + iq q λ ω ω (cid:19) . (15)2. Negative norm, positive energy . The vacuum is now defined as a | i = 0 and a | i =0 . Using p i = − i∂/∂q i one obtains the ground-state wave function ψ ( q , q ) ∝ exp (cid:18) − q ω ω + q /λ ω + ω ) − iq q λ ω ω (cid:19) . (16)If λ = 1 the situation is bad, as emphasized by [12]: the positive-norm quantization gives anormalizable wave-function ψ ˜0 but negative energies; the negative-norm quantization gives aground state wave function not normalizable in q = ˙ q . Excited states have the same problem.However, as we will show in section 4, consistency demands the negative-norm Dirac-Paulirepresentation of a canonical coordinate which roughly amounts to choosing an imaginary λ ,e.g. λ = − i . One then obtains positive energy, negative norm, and a wave function ψ ( q , q ) normalizable in q and q = − i ˙ q . As we will now discuss, despite the strange i factor, ˙ q = iq as well as the Ostrogradski Hamiltonian H = iq p + · · · = ˙ qp + · · · are self-adjoint, so thattime evolution is unitary. 5 Quantum mechanics with negative norm
We here discuss quantum mechanics with negative norm from a general point of view. Neg-ative norm states require putting some minus sign here and there. It is convenient to bemore general and consider a Hilbert-like space with generic, possibly negative, constant norm(called Krein space by mathematicians) and develop a basis-independent formalism. This willlet us to clarify confusions, in particular about self-adjoint operators that are represented (insome basis) by non-hermitian matrices, allowing us to understand the unusual imaginary λ introduced in the previous section.We follow the notations used in general relativity, rewriting the quantum state metricas h n | m i = η nm and defining the inverse metric ( η ) nm ≡ ( η − ) nm , the contro-variant ket | n i = η nm | m i such that h n | m i = η nm and h n | m i = δ nm = h n | m i . Summations over repeatedindexes are implicit. As usual, bras denote complex conjugate of kets.A generic state | ψ i can be expanded in either the ‘covariant’ or the ‘controvariant’ basis: ψ n ≡ h n | ψ i , ψ n ≡ h n | ψ i . (17)Then | ψ i = ψ n | n i = ψ n | n i . (18)A generic linear operator A can be written as a matrix in 4 different ways: A nm ≡ h n | A | m i , A nm ≡ h n | A | m i , A nm ≡ h n | A | m i , A nm ≡ h n | A | m i . (19)Then A = A nm | n ih m | = A nm | n ih m | = A nm | n ih m | = A nm | n ih m | . (20)The components of the matrices are related by A nm = η nn ′ A n ′ m ′ η m ′ m which is an iso-spectral transformation: the eigenvalues do not change because the matrix A gets left-multiplied by η and right-multiplied by its inverse.The unity operator is represented by nm = η nm and nm = η nm and expanded as η nm | n ih m | = η nm | n ih m | = | n ih n | = | n ih n | . (21)Operator multiplication becomes, in components, ( AB ) nm = A nn ′ η n ′ m ′ B m ′ m . Expectationvalues are given by h ψ | A | ψ i / h ψ | ψ i .The adjoint A † of an operator A is defined, as usual, as the operator such that | ψ ′ i = A | ψ i implies h ψ ′ | = h ψ | A † . Thereby for generic matrix elements one has h ψ | A † | ψ i ≡ h ψ | A | ψ i ∗ ,and the relation for the components ( A † ) nm = A ∗ mn i.e. ( A † ) nm = A mn ∗ i.e. ( A † ) nm = ( A mn ) ∗ . (22)The covariant components of a self-adjoint operator A satisfy the usual condition: a self-adjoint operator is described by a hermitian matrix, A nm . The same result holds for the6ontro-variant matrix A nm . The mixed components satisfy a different condition, where com-plex conjugation is supplemented by an iso-spectral transformation: A nm = ( ηA ∗ T η − ) nm . A self-adjoint operator, A † = A has real expectation values h ψ | A | ψ i / h ψ | ψ i , although thematrix A mn that represents it can be anti-hermitian.The mixed components directly enter into the eigenvector equation A | ψ i = A ψ | ψ i : A nm ψ m = A nm ′ η m ′ m ψ m = A ψ ψ n or A nm ψ m = η nn ′ A n ′ m ψ m = A ψ ψ n (23)where A ψ is the eigenvalue. Let us now consider a self-adjoint operator H (later it will be theHamiltonian), with eigenstates | E n i and eigenvalues E n . The identity h E n | H | E m i = h E n | E m i E m = E ∗ n h E n | E m i (24)tells that H can have three different kinds of eigenstates: + ) orthogonal eigenstates h E n | E m i = 0 with real E n and norm h E n | E n i = +1 ; − ) orthogonal eigenstates h E n | E m i = 0 with real E n and norm h E n | E n i = − ;0) pairs of complex conjugated eigenvalues, E n = E ∗ m with h E n | E m i 6 = 0 and zero norm, h E n | E n i = 0 .In the classical analogue, the latter possibility corresponds to a ghost which is also a tachyon,which is a different kind of instability, to be avoided even in absence of ghosts.It is often convenient to choose a basis of eigenstates of H : | n i = | E n i . The associatedcontro-variant states | n i then satisfy H | n i = E ∗ n | n i . In this basis the space splits into twosectors: positive norm and negative norm, plus the possible pairs of zero-norm states. Thetwo sectors experience a joint dynamics only if the initial state has a quantum entanglementamong them. The evolution equation i∂ t | ψ i = H | ψ i becomes i ∂∂t ψ n = H nm η mm ′ ψ m ′ or i ∂∂t ψ n = η nn ′ H n ′ m ψ m . (25)The norm of any state | ψ ( t ) i is conserved by time evolution if H is self-adjoint: i ∂∂t h ψ ′ ( t ) | ψ ( t ) i = h ψ ′ | H − H † | ψ i = 0 . (26) Here T denotes the matrix transpose; ∗ denotes complex conjugation; † denotes the adjoint operation, thatgeneralizes the usual hermitian conjugation and reduces to it in the positive norm case. We never use † todenote hermitian conjugation of a matrix. H leads to unitary time evolution. The explicit solution can bewritten as | ψ ( t ) i = U ( t ) | ψ (0) i with U ( t ) = T e − i R H ( t ) dt , where T is the usual time-ordering.In components, ψ n ( t ) = U nm ψ m (0) = U nm ′ η m ′ m ψ m ′ (0) or ψ n ( t ) = U nm ψ m (0) = η nn ′ U n ′ m ψ m (0) . (27)Having written generic-metric quantum mechanics in an abstract formalism that resem-bles as much as possible the usual positive-norm formalism, let us now emphasize the keydifferences. For simplicity, let us consider a time-independent H . One can then expand U = e − iHt = P ∞ n =0 ( − iHt ) n /n ! . • Writing U in mixed components, U nm is the naive exponentiation of the matrix H nm .However, the mixed components of a self-adjoint H do not form a hermitian matrix.Rather, the self-adjoint condition in eq. (22) dictates that they are hermitian up to aniso-spectral transformation. • The covariant components of a self-adjoint H satisfy the usual Hermiticity H ∗ nm = H mn .However, the covariant components U nm are not given by the naive matrix exponentia-tion of H nm . Rather, extra metric factors appear to covariantize the expansion: U nm = η nm + η nn ′ ( − iHt ) n ′ m ′ η m ′ m + 12 η nn ′ ( − iHt ) n ′ r ′ η r ′ s ′ ( − iHt ) s ′ m ′ η m ′ m + · · · (28)Correspondingly, the unitarity condition U † U = 1 written in covariant components is U ∗ n ′ n η n ′ m ′ U m ′ m = η nm , while in mixed components one gets the usual U k ∗ n U km = δ nm .Practical computations often employ perturbation theory, which can now be easily gener-alized to generic norm. Decomposing H = H + V ( t ) , the I nteraction picture is related tothe Schroedinger picture as A I = e iH t Ae − iH t where A is any operator (including V ). Timeevolution is given by U I ( t i , t f ) = T e − i R tfti dt V I ( t ) = 1 − i Z t f t i dt ′ V I ( t ′ ) − Z t f t i dt ′ Z t ′ t i dt ′′ V I ( t ′ ) V I ( t ′′ ) + · · · . (29)The above explicit form of U I shows that the energy conserved by quantum evolution (up tothe usual quantum uncertainty ∆ t ∆ E ≥ ¯ h ) are the eigenvalues of H . Let us consider forexample a time-independent interaction V and an initial state and a final state which areenergy eigenstates with eigenvalues E i and E f . Defining V f i = h f | V | i i , at first order one has |h f | U | i i| ≃ (cid:12)(cid:12)(cid:12)(cid:12) Z t dt ′ e i ( E f − E i ) t ′ V f i (cid:12)(cid:12)(cid:12)(cid:12) = 4 | V f i | | E f − E i | sin ( E f − E i ) t t →∞ ≃ πt | V f i | δ ( E f − E i ) . (30)This means that energy conservation reads E f = E i , up to the usual quantum uncertainty / ( t f − t i ) . Higher order corrections give the usual sum over intermediate quasi-on-shellstates. 8 .2 Example: the indefinite-norm two-state system Let us consider a two-state system: | + i with positive unit norm, and | − i with negative unitnorm. Without loss of generality, by redefining the relative phase of the two states and addinga constant overall energy, one can trivially write the most generic self-adjoint Hamiltonian as H = 12 (cid:18) | + i | − ih + | E R − iE I h − | iE I E R (cid:19) = 12 (cid:18) | + i | − ih + | E R iE I h − | iE I − E R (cid:19) (31)having used | ± i = ±| ± i . We see that the H nm components are hermitian, unlike the H nm components. The eigenvalues of H are E ± = ± E with E = p E R − E I / . The correspondingeigenstates are | E + i = r γ + 12 | + i − i r γ − | − i , | E − i = i r γ − | + i + r γ + 12 | − i . (32)where γ = 1 / p − E I /E R is a ‘boost factor’ that substitutes the usual mixing angle. • If E I < E R the eigenvalues of H are real, the orthogonal eigenvectors satisfy h E ± | E ± i = ± , and tend to get closer to the ‘light-cone’ of zero-norm states as E I increases. Thecomponents of U = e − iHt oscillate in time: U = (cid:18) | + i | − ih + | cos( Et ) − iγ sin( Et ) p γ − Et ) h − | p γ − Et ) cos( Et ) + iγ sin( Et ) (cid:19) . (33)The unusual feature is that |h ± | U | ± i| oscillates between 1 and γ ≥ . • In the critical case, E R = E I , such that γ = ∞ , the two eigenstates become degeneratewith energy E = 0 . The two eigenvectors also become degenerate, and parallel to thezero-norm state ∝ | + i + i | − i . The evolution operator is U = (cid:18) | + i | − ih + | − iE R t/ E R t/ h − | E R t/ iE R t/ (cid:19) . (34)This exemplifies a more general result: zero-norm eigenstates with complex eigenvaluesappear when, increasing the interaction, a level crossing between a positive-norm anda negative-norm eigenstate takes place; the Hamiltonian becomes degenerate at thecritical transition. • If the interaction E I is strong enough, E I > E R , one has a pair of complex conju-gated eigenvalues, with zero-norm eigenvectors that satisfy h E + | E − i = 1 and describetachyonic ghosts: their time-evolution factor e − iE ± t also contains a real exponential,9n analogy to tachyonic states present in positive-norm theories. In the extreme limit E I ≫ E R the eigenvalues of H are ± iE I / , and the time evolution operator is: U = (cid:18) | + i | − ih + | cosh( E I t/
2) sinh( E I t/ h − | sinh( E I t/
2) cosh( E I t/ (cid:19) . (35)This runaway happens whenever H has a pair of complex eigenvalues E + = E ∗− , as clearwriting time evolution in terms of energy eigenstates, | ψ ( t ) i = ψ E + e − iE + t | E + i + ψ E − e − iE − t | E − i .Both the norm of | ψ ( t ) i and the real expectation value of H are preserved by time evolution: h ψ ( t ) | H | ψ ( t ) ih ψ ( t ) | ψ ( t ) i = E + ψ E + ψ E − ∗ + c.c. ψ E + ψ E − ∗ + c.c. . (36) We here study the concrete system that lies at the basis of perturbative Quantum Field The-ory: the harmonic oscillator. As discussed by Lee and Wick [17] it admits two inequivalentquantizations: positive norm, and indefinite norm.Let us first recall the standard oscillator, described (up to irrelevant constants) by theHamiltonian H = ( p + q ) with [ q, p ] = i . Defining a = q + ip √ , a † = q − ip √ (37)one has [ a, a † ] = 1 and H = ( aa † + a † a ) / .Let us next consider a more general system described by the following Hamiltonian andcommutation relations: H = s H a † a + aa † , [ a, a † ] = s. (38)For s = s H = 1 this reduces to the usual oscillator. We now show that s = s H = − definesanother consistent positive-energy theory. The symbol a † here indicates the adjoint of a ,which generalizes the Hermitian conjugate to negative norm.We again define the vacuum as a | i = 0 and the excited states as | n i = a † | n − i / √ n =( a † ) n | i / √ n ! . Its inverse is a | n i = s √ n | n − i . The state metric is η nm ≡ h m | n i = s n δ nm . Thenorm is determined by the dynamics, and odd states have negative norm for s = − . Theinverse metric is η nm = s − n δ nm and the contro-variant states are | n i = s − n | n i . In componentsone has: a = | i | i | i | i · · ·h | s · · ·h | √ s · · ·h | √ s · · ·h | · · · ... ... ... ... ... . . . = | i | i | i | i · · ·h | · · ·h | √ · · ·h | √ · · ·h | · · · ... ... ... ... ... . . . (39) † = | i | i | i | i · · ·h | · · ·h | s · · ·h | √ s · · ·h | √ s · · · ... ... ... ... ... . . . = s | i | i | i | i · · ·h | · · ·h | · · ·h | √ · · ·h | √ · · · ... ... ... ... ... . . . (40) In components the commutation relations read [ a, a † ] nm = ( a · η · a † − a † · η · a ) nm = s n +1 δ nm = sη nm i.e. [ a, a † ] = s X n | n ih n | = s (41)and the Hamiltonian is: H nm = ( n + 12 ) s H s n +1 δ nm = E n η nm i.e. H = ∞ X n =0 E n | n ih n | (42)where E n = ( n + ) ss H are the Hamiltonian eigenvalues, H | n i = E n | n i . We see that positive-energy eigenvalues are obtained for s = s H = 1 (the usual case with positive H and positivenorm), but also for s = s H = − (negative H and negative norm).Concerning the negative-norm case, s = − , notice that the harmonic oscillator does notpredict tachyonic ghosts with zero norm. Furthermore the matrix elements a nm are not thehermitian conjugates of ( a † ) nm , such that the operators q = ( a + a † ) / √ and p = i ( a † − a ) / √ are represented by matrices q nm and p nm which are not Hermitian. This is why variousauthors who look at these matrices improperly speak of ‘anti-Hermitian’ operators. Neverthe-less, q and p are self-adjoint operators. We will now find their coordinate representation. Starting from the harmonic oscillator, we now describe a more general representation of apair of canonical coordinate variables q, p . Parity flips q → − q and p → − p . In the harmonicoscillator case, this means a → − a and a † → − a † : so eigenstates | n i with even (odd) n areeven (odd) under parity. In the negative norm quantization, states with odd n also havenegative norm. Going to the coordinate wave-function representation (we use the notation x for the coordinate, that later will become field space), this means that the norm is h ψ ′ | ψ i = Z dx [ ψ ′∗ even ( x ) ψ even ( x ) − ψ ′∗ odd ( x ) ψ odd ( x )] = Z dx ψ ′∗ ( x ) ψ ( − x ) . (43)The corresponding unit operator is R dx | − x ih x | . Switching to the formalism appropriatefor generic norm, one has the norm h x ′ | x i = δ ( x + x ′ ) . Thereby the controvariant state is | x i = | − x i and it satisfies the usual h x ′ | x i = δ ( x − x ′ ) . As already discussed around eq. (17),a state can be expanded as | ψ i = R dx ψ ( x ) | x i = R dx ψ ( x ) | x i with ψ ( x ) = h x | ψ i and ψ ( x ) = h x | ψ i = ψ ( − x ) . 11hat is emerging from the harmonic oscillator computation is a more general structure:a coordinate space representation of a pair q, p of conjugated canonical variables that differsfrom the usual positive-norm representation q | x i = x | x i , p | x i = + i ddx | x i (44)which implies h x | p | ψ i = ( − id/dx ) ψ ( x ) so that it satisfies h x | [ q, p ] | ψ i = i h x | ψ i .The negative-norm coordinate representation, originally discussed by Dirac [13] andPauli [14], is q | x i = ix | x i , p | x i = + ddx | x i . (45)Although q looks anti-hermitian, taking into account the extra i as well as the negative norm,these unusual features combine to form a self-adjoint q : h x ′ | q † | x i = h x | q | x ′ i ∗ = [ ix ′ δ ( x + x ′ )] ∗ = ixδ ( x + x ′ ) = h x ′ | q | x i . (46)This means that h ψ | q | ψ i = R dq ψ ∗ ( − q ) iq ψ ( q ) is real. A similar result holds for p . Whenacting on wave-functions one has h x | q | ψ i = − ixψ ( x ) and h x | p | ψ i = (+ d/dx ) ψ ( x ) , giving thedesired [ q, p ] = i commutator. Defining momentum eigenstates as p | p i = ip | p i one finds h q | p i = e ipq / √ π , h p ′ | p i = δ ( p + p ′ ) . The operator q acts as h q | q | p i = ( − d/dp ) h q | p i . One canagain define | p i = | − p i such that R dp | p ih p | .The i factor that differentiates the usual representation from the Dirac-Pauli representa-tion has an impact on the time-inversion parity. As usual, a positive energy spectrum demandsthat the time inversion symmetry is anti-unitary. Then, in the Dirac-Pauli quantization q isnaturally T -odd and p is naturally T -even (while the opposite holds in the usual quantization,unless T is defined adding ad-hoc extra signs). This will play a key role in section 4.We are now ready to come back to the harmonic oscillator. Inserting into the condi-tion h x | a | i = 0 the standard positive-norm representation such that a = ( q + ip ) / √ x + s d/dx ) / √ gives a differential equation which implies the ground-state wave function ψ ( x ) ∝ e − sx / . This is normalizable for s = 1 (positive norm) and non-normalizable for s = − , where s was defined in eq. (38). This problem was emphaized e.g. by Woodard [12]that thereby dismissed the negative norm quantization as purely formal. However, theproblem arises because the positive-norm representation of q, p was used together with thenegative-norm oscillator: the problem is just a manifestation of the inconsistency of the as-sumptions. Consistency demands that the negative norm harmonic oscillator must be accom-panied by the negative-norm Dirac-Pauli coordinate space representation of the self-adjoint q, p operators, eq. (45). Then, the condition h x | a | i = 0 leads to a normalizable wave func-tion for the ground state ψ ∝ e − x / , as well as for the excited states. The Dirac-Pauli choicethereby provides a self-consistent description of the negative-norm oscillator. Furthermore,as discussed in the next section, in the 4-derivative case the Pauli-Dirac representation isdemanded by simple considerations. 12orm h x | q | ψ i T -parity h x | p | ψ i T -parity harmonic oscillator with E > positive xψ ( x ) even − i dψ/dx odd ψ ( q ) ∝ e − q / and H = + ( q + p ) indefinite − ixψ ( x ) odd dψ/dx even ψ ( q ) ∝ e − q / and H = − ( q + p ) Table 1:
Coordinate representations of a pair of canonical variables [ q, p ] = i , and the associatedground-state wave functions for the positive-energy harmonic oscillator. As discussed in the previous section, and as summarized in table 1, quantum mechanics hastwo faces: a canonical coordinate can be representedi) in the usual way with positive norm;ii) in the Dirac-Pauli way, with negative norm, eq. (45).As we now show, theories with 4-derivatives want this latter quantization choice (that, in thegravitational case, corresponds to a renormalizable theory with positive energy).A single 4-derivative real coordinate q ( t ) contains two degrees of freedom. The Ostro-gradski procedure (section 2) rewrites the theory as a Hamiltonian system of two canonicalcoordinates, q = q and q = λ ˙ q . The key new feature that arises in 4-derivative theories isthat ˙ q becomes an extra canonical coordinate. In the classical theory q is just an auxiliaryvariable, and λ is an irrelevant constant: Ostrogradski used λ = 1 .In the quantum theory, q and q are operators that allow to define the basis | q , q i . Wenow show that the usual quantization must be used for q and that the Dirac-Pauli quanti-zation must be chosen for q , which is equivalent to (and more transparent than) fixing animaginary λ and using the canonical representation.As usual, the operator q = q is invariant under time-reversal t → − t , and thereby it canfollow the usual T -even representation. On the other hand, the operator ˙ q is T -odd, becauseof the time derivative: the time-inversion operator T transforms it as T ˙ qT − = − ˙ q . This isthe novel key feature.Taking into account that T is anti-unitary, one can equivalently define a usual T -evencoordinate q = λ ˙ q by choosing an imaginary λ . However, it is simpler to forget the λ factors and just declare that the self-adjoint operator ˙ q is T -odd and thereby it follows the Alternative routes lead to the same conclusion. For example, one can use the T -even ¨ q (instead of ˙ q )as second canonical coordinate. In the Ostrogradski formalism ¨ q = − p is a momentum. So again one getsa canonical coordinate with unusual T -parity (normally a momentum is T -odd). In general, the invarianceof the commutation relation [ q , p ] = i under the anti-unitary time-inversion implies that q and p haveopposite T -parities. One can switch q ↔ p in order to restore their usual T -parities: but their commutatorchanges sign, implying again negative norm quantization. This is indeed what happens in the auxiliary variableformalism, used in various forms in the literature as an alternative to the canonical Ostrogradski formalism (seee.g. [22, 24, 29]). This formalism is convenient when dealing with quantum field theory instead of quantummechanics with a finite number of degrees of freedom. In order to facilitate the contact, we summarize the -odd Pauli-Dirac representation. Then, the Ostrogradski Hamiltonian of eq. (6) is T -even.The states satisfy T | q, ˙ q i = | q, ˙ q i since ˙ q has imaginary eigenvalues and since T is anti-unitary.The strange extra factor of i has been justified from first principles. A posteriori, it wasnot so strange. After all, it is well known that the self-adjoint spatial gradient is i ~ ∇ ratherthan ~ ∇ . In a relativistic theory, one could have guessed that similarly the self-adjoint timederivative is i∂/∂t rather than ∂/∂t . Loosely speaking, while from a classical perspective ˙ q was the natural auxiliary variable, from a quantum perspective the natural extra coordinateoperator is i ˙ q . Using the Heisenberg representation, one has q ( t ) = U † ( t ) q (0) U ( t ) and ˙ q = − i [ q, H ] = U † ( t ) ˙ q (0) U ( t ) with unitary U , so q ( t ) keeps real eigenvalues and ˙ q ( t ) keeps imaginary eigen-values at any t (these statements are not contradictory, given that q ( t ) also depends on p (0) and p (0) ). auxiliary variable formalism below. Restarting from the Lagrangian in eq. (3), we add zero as a perfect squarecontaining an auxiliary variable a : L = L + 12 (cid:20) ¨ q + ( ω + ω ) q − a (cid:21) . (47)Expanding the square cancels both the second-order and the fourth-order kinetic terms for q leaving L = − a ¨ q ω − ω ) q − ( ω + ω ) aq a − V ( q ) . (48)Going to the free theory V = 0 , we can diagonalize the kinetic and mass term through the field redefinition (cid:26) a = p ω − ω (˜ q − ˜ q ) q = (˜ q + ˜ q ) / p ω − ω i.e. ˜ q , = q q ω − ω ∓ a p ω − ω (49)obtaining two decoupled oscillators L = ˙˜ q − ω ˜ q − ˙˜ q − ω ˜ q . (50)From its classical solution, a = 2¨ q + ( ω + ω ) q , we see that a roughly corresponds to the Ostrogradski p .Furthermore, inserting such classical solution in eq. (49) one recovers the formalism used in [22] ˜ q = ¨ q + ω q p ω − ω , ˜ q = − ¨ q + ω q p ω − ω . (51) The possibility of converting a non-normalizable wave-function ψ ∝ e z / into a normalizable one byrestricting z = x + iy to the imaginary axis, rather than along the real axis, was presented as an ad hoc recipeto get something sensible in earlier works by Bender and Mannheim [25]. At the technical level, their approachdiffers from ours because they added an i factor to the variable q , rather than to ˙ q . Our approach follows fromgeneral considerations, and has the advantage that q, p and thereby the Hamiltonian are self-adjoint (althoughtheir matrix representations look anti-hermitian), such that the generalization to an interacting theory will beimmediate (section 6.1). he frequency eigenstates We conclude this section by computing what the Dirac-Pauli representation adopted for q = ˙ q implies for the frequency eigenstates. We restart from the Hamiltonian eq. (6) and bring it indiagonal form H = −
12 (˜ p ˜ λ + ω ˜ q ˜ λ ) + 12 (˜ p + ω ˜ q ) (52)through the canonical transformation q = ˜ q − ˜ λ ˜ p /ω p ω − ω , q λ = ˜ p − ω ˜ q / ˜ λ p ω − ω , p = ω ω ˜ p − ω ˜ q / ˜ λ p ω − ω , p λ = ω ˜ q − ω ˜ λ ˜ p p ω − ω . (53)which satisfies q p − q p = ˜ p ˜ q − ˜ p ˜ q . For the sake of generality, we here allow for genericfactors λ and ˜ λ . The non-vanishing commutators, [˜ q , ˜ p ] = i and [˜ q , ˜ p ] = i , can be rewrittenin terms of ˜ a = p ω / q + i ˜ p /ω ) and of ˜ a = p ω / q / ˜ λ − i ˜ λ ˜ p /ω ) reproducing theHamiltonian of eq. (14) and the commutators of eq. (13). The ground-state wave function iseasily computed imposing h ˜ q , ˜ q | ˜ a , | i = 0 finding ψ (˜ q , ˜ q ) ∝ exp (cid:20) − ω ˜ q ω ˜ q λ (cid:21) . (54)For ˜ λ = 1 it is not normalizable [26]. It is normalizable if instead | Im ˜ λ | > | Re ˜ λ | .The Dirac-Pauli representation for q , p corresponds to imaginary λ . Imposing that q , p are T -odd and that q , p are T -even (i.e. that q and p have the unusual T parity) impliesthat ˜ q , ˜ p are T -odd and that ˜ q , ˜ p are T -even (i.e. that the canonical coordinates of thenegative norm mode ˜ q and ˜ p have the unusual T parity). This is obtained for imaginary ˜ λ .As a check, let us connect the q , q basis with the ˜ q , ˜ q basis for generic λ and ˜ λ . It isconvenient to start from the T -odd basis ˜ q , ˜ p , in which the ground state wave function is ψ (˜ q , ˜ p ) ∝ exp (cid:20) − ˜ p ω + ω ˜ q λ (cid:21) . (55)Next, the transition to the T -odd variables p , q is simply h p , q | ˜ q , ˜ p i ∝ δ (cid:18) ˜ q ˜ λ − p − q ω /λω p ω − ω (cid:19) δ (cid:18) ˜ p − p − q ω /λ p ω − ω (cid:19) . (56)Inserting the change of variables dictated by the δ functions into ψ (˜ q , ˜ p ) one obtains ψ ( p , q ) ∝ exp (cid:20) − p + 2 ω ω p q /λ − ω ω ( ω + ω ω + ω )( q /λ ) ω ω ( ω + ω ) (cid:21) (57)where q and p are both complex and linked by Re p = ω Re ( q /λ ) and Im p = − ω Im ( q /λ ) . ψ can be trivially analytically continued to real p , q . For λ = ± i it remains a bounded Gaus-sian. Finally, one performs the Fourier transform from p to q , obtaining from ψ ( p , q ) theground state wave function ψ ( q , q ) , which agrees with eq. (16). The same equality holdsfor excited states, that can be computed acting with creation operators on the ground state.In the limit ω = ω one gets the critical situation described in section 3.2.15 Path-integral quantization
We now present the path-integral quantization of the same 4-derivative theory.
Our generic-norm formalism makes easy to write down the equivalent path-integral formal-ism, an issue already considered in [21]. Inserting R dq | q ih q | at intermediate times t m = t i + m dt one has h q f ,t f | q i ,t i i = Y m Z dq m h q m +1 ,t m +1 | q m ,t m i . (58)Each step h q m +1 ,t m +1 | q m ,t m i can be evaluated as h q m +1 | e − iHdt | q m i = Z dp m h q m +1 | p m ih p m | e − iHdt | q m i = Z dp m π e i [ p m ( q m +1 − q m ) − H cl dt ] (59)having defined H cl ≡ h p | H | q ih p | q i (60)and used h q | p i = e ipq / √ π and h p | q i = e − ipq / √ π . The final result is the path integral h q f ,t f | q i ,t i i = Z Dq Dp e i R dt [ p ˙ q − H cl ] where Dq Dp = lim dt → Y m dq m dp m π (61)and with boundary conditions q ( t i ) = q i , q ( t f ) = q f . Applying the generic path-integral of eq. (61) to the 4-derivative oscillator in the canonicalOstrogradski formalism, one gets the transition amplitude h q f ,q f ,t f | q i ,q i ,t i i ∝ Z Dq Dp Dq Dp exp (cid:20) i Z dt [ p ˙ q + p ˙ q − H cl + J q + J q ] (cid:21) (62)where for generality we added currents J , (such that acting with functional derivatives withrespect to them one can form more general matrix elements of time-ordered operators; J is T -even and J is T -odd). The Pauli-Dirac representation for ˙ q manifests in two ways:1) A propagator with an unusual − in its external state.Rewriting the transition amplitude in the usual positive-norm formalism, it acquires an usual − sign, becoming h q f , − ˙ q f , t f | q i , ˙ q i , t i i . In the limit t f → t i one has h q f , ˙ q f | q i , ˙ q i i = δ ( q f − q i ) δ ( ˙ q f − q i ) , so that the unusual − sign is equivalent to the Dirac-Pauli negative norm. Furthermore,the T -odd nature of ˙ q is hardwired in the path-integral, as a geometrical feature. For eachtrajectory q ( t ) with boundary conditions q ( t i,f ) = q i,f and ˙ q ( t i,f ) = ˙ q i,f the time-invertedtrajectory has the same action and the following boundary conditions: q i → q f , q f → q i , ˙ q i → − ˙ q f , ˙ q f → − ˙ q i . (64)Thereby the propagator given by the path-integral satisfies the identity h q f , − ˙ q f , t f | q i , ˙ q i , t i i = h q i , ˙ q i , t f | q f , − ˙ q f , t i i (65)which is equivalent to the operator identity h ψ f | ψ i i = h T ψ i | T ψ f i given that T | q, ˙ q, t i = | q, ˙ q, − t i .2) An unusual classical Hamiltonian.Inserting the Ostrogradski Hamiltonian of eq. (6) in the generic path integral of eq. (61) onegets the following classical Hamiltonian: H cl = h p ,p | H | q ,q ih p ,p | q ,q i = ip q + p ω + ω q + ω ω q + V ( q ) . (66)This is the same as eq. (6) with λ = − i . H cl can be complex because q , p , in the Dirac-Paulirepresentation, have complex eigenvalues. Thanks to the unusual i , it is invariant undertime-reversal.Let us now try to evaluate the path-integral. As usual, one can perform the Gaussian Dp Dp path integrals. The Dp path-integral formally gives the Dirac delta function δ ( q − λ ˙ q ) , allowing to eliminate the Dq path-integral, leaving h q f , ˙ q f ,t f | q i , ˙ q i ,t i i ∝ Z Dq exp (cid:20) i Z dt [ L ( q ) + J q + J λ ˙ q ] (cid:21) , (67)where L coincides with the original 4-derivative Lagrangian. By partial integration, thesource term for ˙ q can be transformed into a source for q or for ¨ q (like in the auxiliary-fieldmethod). This computation however has three problems:1. the Dp path-integral is, in general, divergent. Thereby the subsequent result is onlyformal. In the limit dt = t f − t i → the classical action becomes S cl ≃ dt (cid:18) q f − q i − dt ˙ q i + ˙ q f (cid:19) + ( ˙ q f − ˙ q i ) dt + · · · (63)which is minimal for a motion with constant speed ( ˙ q i + ˙ q f ) / . The classical action satisfies S ( q f , ˙ q f , t f ; q i , ˙ q i , t i ) = − S ( q i , ˙ q i , t i ; q f , ˙ q f , t f ) as well as S cl ( q f , ˙ q f , t f ; q i , ˙ q i , t i ) = − S cl ( q f , − ˙ q f , t i ; q f , − ˙ q i , t i ) . The classical Hamiltonian is real if one instead uses the equivalent oscillator basis ˜ q , ˜ q , ˜ p , ˜ p of eq. (53).
17. the δ ( q − λ ˙ q ) always vanishes if q and q are real. Thereby the Dq path-integral isonly formal.3. Once interactions are turned on, the Lagrangian admits classical run-away solutions,reflected in the path-integral.Given that the theory is well defined in the operator formalism, somehow this path integralmust have a sense. A sensible path-integral is found by restarting from eq. (62) and continuing it to Euclideantime, it = t E , such that dq/dt = i dq/dt E i.e. ˙ q = iq ′ . One gets the Euclidean path integral h q f ,q f ,t Ef | q i ,q i ,t Ei i ∝ Z Dq Dq Dp Dp exp (cid:20) Z dt E ( ip q ′ + ip q ′ − H cl + J q + J q ) (cid:21) . (68)Now the Dp integral is convergent and gives δ ( q − q ′ ) , such that the Dq path integral justfixes q = q ′ . Next, the remaining terms in H cl are a sum of positive squares so all otherintegrals are convergent. Performing them one finds the Lagrangian Euclidean path-integral: h q f ,q ′ f ,t Ef | q i ,q ′ i ,t Ei i ∝ Z Dq exp (cid:20) − Z dt E [ L E ( q ) + J q + J q ′ ] (cid:21) (69)where the classical Euclidean Lagrangian corresponding to eq. (3) is L E = 12 (cid:18) d qdt E (cid:19) + ω + ω (cid:18) dqdt E (cid:19) + ω ω q + V ( q ) . (70)Let us now check the result. The classical free solution is q ( t E ) = a e − ω t E + a e − ω t E + b e ω t E + b e ω t E . (71)It already contains run-away exponentials, characteristic of any Euclidean theory. Interac-tions compatible with the positivity of the action lead to an equally good path-integral. Byimposing the boundary conditions q = q ′ = 0 at t Ei = −∞ and evaluating the classical action,one finds the normalizable ground-state wave function h q, q ′ , t E = 0 | , , t E = −∞i ∝ exp (cid:20) − q ω ω + q ′ ω + ω ) + qq ′ ω ω (cid:21) . (72)This agrees with the ground-state wave-function ψ ( q , q ) in eq. (16), that was computed inthe Dirac-Pauli formalism in Minkowski space, after identifying q = q and q ′ = q . In otherwords, q ′ = dq/dt E = − idq/dt coincides with q , as computed for λ = − i . The novel featureintroduced by 4-derivatives is that q ′ must not be continued into an imaginary − i ˙ q (whichwould give divergent wave functions), because it already describes the T -odd variable q ,which contains the Dirac-Pauli i factor of eq. (45). The final result is that the Minkowskiantheory is an unusual analytic continuation of the Euclidean theory. Hawking and Hertog [22] found a non-normalizable Minkowskian wave-function because they expressed Interactions, Quantum Field Theory, probability
Summarising, we so far considered a 4-derivative harmonic oscillator. One might think thatwe achieved nothing [26]. After all, a classical 4-derivative harmonic oscillator has no run-away problems, see eq. (9), given that it splits into two decoupled oscillators, one with neg-ative energy and one with positive energy. The classical trouble starts when they interact. Inthis section we will explain that we have achieved instead something useful in an interactingquantum field theory.
The quantum formalism was so far developed for the harmonic oscillator (which correspondsto the modes of a free 4-derivative quantum field theory), finding that the quantum theory hasa positive energy spectrum and no run-away behaviours. Adding interactions, the quantuminteracting inherits all these good properties, as long as interactions are perturbative and aslong as the interacting Hamiltonian H remains self-adjoint.The second issue was the main obstacle to past attempts of adding ad-hoc unusual i factorsin order to make the quantum free theory consistent [25] (normalizable wave functions andunitary evolution with negative norm and positive energy eigenvalues): adding extra complexfactors can render interactions complex, ruining the theory [26].In our approach the only extra i factor arose from a principled reason: ˙ q is a T -oddcoordinate that follows the negative-norm Dirac-Pauli representation. This satisfies all theproperties of quantum mechanics, as generalised to negative norms: ˙ q itself is self-adjoint,like q and ¨ q . Thereby any interaction which is a real function of them is self-adjoint. Ourprocedure immediately generalizes to the interacting case (in agravity [8] all interactions aredictated by general covariance).The perturbativity assumption means that, as long as the energy spectrum of the freeoscillator gets slightly distorted by interactions, the energy eigenvalues will remain real andbounded from below (strongly interacting theories could also be good; however they seemnot needed for the physical application to agravity [8]).One might worry that, even if all energy eigenvalues are positive, the theory possessesnegative-norm states with h ψ | H | ψ i < . Eq. (29) shows how transition amplitudes can becomputed trough perturbation theory: we see that the energy eigenvalues are the quantitythat enters into conservation of energy. Thereby a theory where all eigenvalues of H (of H in the perturbative expansion) are positive is consistent. As usual, perturbative computationscan be systematised in terms of the propagator. By expressing q = q in terms of the annihi-lation and creation operators a i , a † i through eq.s (11) and (12) and using the commutation eq. (72) in terms of ˙ q , which is not the canonical coordinate q appropriate for 4-derivative theories. Theirproposal of integrating out ˙ q in the Euclidean before performing the analytic continuation to the Minkowskianis not necessary: the Minkowskian wave functions are normalizable if the appropriate analytic continuation isperformed. [˜ a i , ˜ a † i ] = s i we find the propagator h | T q ( t ) q ( t ′ ) | i = h | θ ( t − t ′ ) q ( t ) q ( t ′ ) + θ ( t ′ − t ) q ( t ′ ) q ( t ) | i (73a) = 1 ω − ω X i s i ω i [ e iω i ( t − t ′ ) θ ( t ′ − t ) + e iω i ( t ′ − t ) θ ( t − t ′ )] (73b) = i ω − ω Z dE π X i s i e − iE ( t − t ′ ) E − ω i + iǫ (73c) = Z dE π − i e − iE ( t − t ′ ) ( E − ω + iǫ )( E − ω + iǫ ) . (73d)where ǫ is a small positive quantity and we used s = − and s = 1 in the last step.One might worry that, using the Heisenberg picture, operators satisfy the time evolutionequation ˙ A = − i [ A, H ] , which looks dangerously similar to the classical equation of motion,as given by Poisson parentheses, which has run-away solutions. However, the quantum so-lutions are equal to the classical solutions only in a free theory. In general operators are notnumbers, and the difference (in particular, the Pauli-Dirac representation) manifests whennon-linear interactions are present. As well known, the Heisenberg equations are in generalsolved by A ( t ) = U † ( t ) A (0) U ( t ) . So, all good properties of negative norm states found in theSchr¨odinger picture remain valid in the Heisenberg picture, given that they are equivalent. As well known, a single harmonic-oscillator degree of freedom q ( t ) is the building block fora field such as φ ( t, x, y, z ) or g µν ( t, x, y, z ) . The expansion of a field in Fourier modes withgiven momentum works in the 4-derivative case similarly to the 2-derivative case. As longas, at the end, we are only interested in S -matrix elements, all the detailed structure of thequantum mechanical theory, such as the wave-functions, gets hidden behind the commutationrelations of eq. (13), which hold separately for each mode. The usual iǫ prescription for thefield propagator dictates that amplitudes can be analytically continued from the Euclidean.Details will be presented elsewhere.One would like to claim that quantum field theory inherits all good properties of quantummechanics also when negative norms are present. However, while in quantum mechanicsinteractions can easily satisfy the condition that avoids ‘tachionic ghosts’ (namely, the inter-action strength between two opposite-norm states must be smaller than their energy differ-ence as discussed in section 3.2), any interesting quantum field theory leads to situationsthat might violate this condition. The simplest situation where this occurs is the decay of aghost (for example a massive spin 2 graviton at rest), which can be degenerate with a multi-particle state (for example two photons going in opposite directions with energy equal tohalf of the ghost mass), such that the interaction, no matter how small, can be smaller thanthe energy difference. Actually, the ghost is degenerate with an infinite number of similarstates, such that an appropriate limit procedure is needed: in the positive norm case, en-tropic considerations allow to interpret this situation as particle decay. A 4-derivative kinetic20erm Π( p ) = − ( p − m )( p − m ) acquires a positive imaginary part. We will explore if ‘ghostdecay’ can be interpreted like in [24]. So far we carefully avoided talking about probabilities.The theory is unitary in a negative-norm space. Thereby the only remaining difficulty isassigning an interpretation to states that entangle positive norm components with negativenorm components. The Copenhagen interpretation added an extra ingredient external to thedeterministic formalism of quantum mechanics: the Born postulate, according to which: “when an observable corresponding to a self-adjoint operator A is measured in a state | ψ i , the result is an eigenvalue A n of A with probability P n = h ψ | Π n | ψ ih ψ | ψ i where Π n = | n ih n |h n | n i (74) is the projector over the eigenstate | n i of A ”. For positive norms, these P n satisfy the probability rules ≤ P n ≤ and P n P n = 1 ; theaverage value of A satisfies h ψ | A | ψ i / h ψ | ψ i = P n A n P n .At the moment we do not have a satisfactory generalisation to indefinite norm. Evenworse, the Born postulate is unsatisfactory by itself, given that it describes a non-local collapseof the wave-function [33]. In order to make progress, one needs to operate close to theheart of quantum mechanics. As well known this presents fatal risks: physicists tend tobecome philosophers. We conclude by listing some interpretations of quantum mechanics,equivalent to the Copenhagen interpretation, which could lead to a satisfactory indefinitivenorm quantum mechanics.1. Feynman clarified the ontological basis of the Born postulate: it agrees with experi-ments, so ‘shut up and compute’. All experiments have so far been performed withpositive norm states. The negative norm graviton predicted by agravity is beyond thereach of present experiments. On the one hand, this is good because it means thatEinstein’s general relativity is recovered at large distances; on the other hand, however,we do not have experimental guidance. Lee and Wick proposed that the interpretationissue is bypassed, given that in quantum field theory we can only observe asymptoticstates, which are made of positive-norm quanta [17]. The Lee-Wick idea may be appliedto the gravitational theory proposed by Stelle [2], as discussed in [5, 34].2. Any self-adjoint Hamiltonian H gives unitary evolution with respect to many differentnorms, since each energy eigenstate evolves picking just a phase. Defining ghost parity G to be the metrics in the special basis of energy eigenstates and | ψ i = G − | ψ i , a possiblegeneralization of the Born postulate to generic norm is (see also [25]) P n = h ψ | Π n | ψ i where Π n = | n ih n | . (75)21he example of section 3.2 gets converted into normal oscillations with mixing angle sin θ = E I /E R . However, h ψ | A | ψ i is real but does not have a probabilistic interpreta-tion, while h ψ | A | ψ i has a probabilistic interpretation but can be complex.3. Various authors claim that the Born postulate is just an emergent phenomenon (some-how like friction) that follows from the fundamental deterministic equations when ap-plied to complex systems in view of spontaneous decoherence [35].4. Cramer [36] proposed a “transactional interpretation”, claiming that EPR non-localityresults from a cancellation of advanced and retarded waves, in a time-symmetric set-up (see also [37]) inspired by the analogous formulation of classical electro-dynamicsproposed by Dirac and Feynman-Wheeler. The h ψ ′ | ψ i amplitude in the Dirac-Pauli co-ordinate representation supports the interpretation as being the overlap of a wave ψ moving forward in time with a wave ψ ′ moving backwards in time.We plan to further investigate such issues. We presented the quantization of 4-derivative theories, finding that a unique structure emerges.We can summarise it as follows.Quantum mechanics has its usual visible face, where a coordinate operator q is repre-sented as q | x i = x | x i . But quantum mechanics also has a hidden face, where q | x i = ix | x i , asfirst pointed out by Dirac and Pauli. Both q and p of a canonical pair [ q, p ] = i are self-adjointin both representations. The main difference is that the usual representation implies positivenorms and q is naturally even under time reflection T , while the DP representation leads tostates with indefinite norm and to a naturally T -odd q (in view of the i factor and of the factthat T is anti-unitary)The Ostrogradski formulation of a 4-derivative degree of freedom q ( t ) (summarised insection 2) employs two canonical coordinates: q = q and q = ˙ q . For the first time wehave observed that q , which is T -even, naturally follows the usual representation, while q which is T -odd, naturally follows the Dirac-Pauli negative-norm quantization. This leadsto a sensible quantum theory with positive energies and normalizable wave-functions, asdiscussed in section 4.In section 3 we presented a new formalism appropriate for generic-norm quantum me-chanics, introducing ‘covariant’ | n i and ‘contro-variant’ | n i basis states. This clarifies why aself-adjoint linear operator can be represented by a matrix that, in some basis, is not hermi-tian. A self-adjoint Hamiltonian leads to unitary time-evolution, in the sense that the negativenorm is preserved. Given that q , ˙ q , ¨ q , . . . are self-adjoint, a Hamiltonian which is a genericreal function of them is self-adjoint, leading to sensible interacting quantum theory providedthat one avoids tachyons, an observation that was previously overlooked. The usual con-dition that the theory should be free of tachyons is generalised to negative norm quantummechanics. 22n section 5 we presented the path-integral formulation of negative-norm quantum me-chanics. The classical Hamiltonian becomes complex. Another new result of this paper is theproof that the normalizable wave functions found in the operator formalism are recoveredfrom the path-integral after performing naive manipulations over ill-defined objects and/oranalytic continuations. In particular, the version of the path-integral in Euclidean time t E = it is well defined, and reproduces the usual wave functions taking into account that dq/dt E al-ready coincides with the Dirac-Pauli ˙ q .The fact that (1) our approach leads to normalizable wave-functions and (2) these wave-functions can also be deduced from a well-defined Euclidean path-integral clearly show thatthe right quantization for ˙ q is the Dirac-Pauli one.Two issues must be addressed before that these results can be used to obtain a predic-tive renormalizable quantum theory of gravity: generalisation to quantum field theory, andgeneralisation of the Born probabilistic interpretation to negative norms. Acknowledgments
This work was supported by the ERC grant NEO-NAT, by the Spanish Ministry of Economy and Competitivenessunder grant FPA2012-32828, Consolider-CPAN (CSD2007-00042), the grant SEV-2012-0249 of the “Centro deExcelencia Severo Ochoa” Programme and the grant HEPHACOS-S2009/ESP1473 from the C.A. de Madrid.We thank all the colleagues who told us that working with ghosts is equivalent to abandoning physics, andEnrique ´Alvarez, Jos´e R. Espinosa, Antonio Gonz´alez-Arroyo, Martti Raidal, Hardi Veermae, Sergey Sibiryakov,and Guido Altarelli for useful comments and encouragement. This was the content of my last conversation withGuido — the present paper was finalized wondering what Guido would have said.
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