Quantum fields on curved spacetimes and a new look at the Unruh effect
Abstract
We describe a new viewpoint on canonical quantization of linear fields on a general curved background that encompasses and generalizes the standard treatment of canonical QFT given in textbooks. Our method permits the construction of pure states and mixed stated with the same technique. We apply our scheme to the study of Rindler QFT and we present a new derivation of the Unruh effect based on invariance arguments.
Keywords:
Quantization. Quantum fields in curved spacetime. Canonical formalism.:
03.70.+k, 04.62.+v, 11.10.z1 First quantization: commutation relations and their representation
Switching from classical to quantum mechanics relies on a set of recipes that have proven to successfully describe a large variety of physical phenomena, although the mysteries of quantum mechanics and its interpretation persist to date.
In the simple case of a mechanical system having a finite number of degrees of freedom, the timehonored procedure of “canonical quantization” essentially amounts to replacing the classical canonical variables and their Poisson brackets^{1}^{1}1The Poisson brackets of two functions of the canonical coordinates are defined as usual by
(2)  
(4)  
(5) 
In a modern viewpoint the “operators” and in Eq. (4) can be regarded as elements of an abstract Heisenberg algebra rather than operators in a Hilbert space. However, under suitable technical assumptions, the fundamental uniqueness theorem by Stone and von Neumann establishes that, up to unitary equivalence, there exists only one representation of the algebraic canonical commutation relations (4) by operators in a Hilbert space :
(6)  
(7)  
(8) 
where the symbol denotes the identity operator in the Hilbert space . Therefore, the distinction between the abstract Heisenberg algebra and its representations by means of operators in a Hilbert space is unnecessary for the study of finitedimensional quantum mechanical systems.
As an elementary example, consider one pointlike particle in the physical space . Following the probabilistic interpretation of quantum mechanics the natural Hilbert space to be considered is and the commutation relations (4) are represented there by the following operators^{2}^{2}2We do not enter here into the unavoidable technical problems related to the unboundedness of operators; see e.g. Reed and Simon (1980) for an account of that important topic.:
(9) 
The canonical commutation relations can be used to actually build the Hilbert space representation starting from a “fundamental” state. To this end, one introduces the ladder operators Dirac (1958); Messiah (1930) as the following complex combinations of the elements of the Heisenberg algebra:
(10) 
the CCR algebra is rewritten as follows:
(11) 
A particularly relevant Hilbert space realization of the commutation relations (11)
(12) 
(13) 
can be constructed starting from a common eigenvector (the fundamental state) of all the annihilation operators with eigenvalue equal to zero:
(14) 
with An orthonormal basis of is then obtained by repeated application of the creation operators to the fundamental state:
(15) 
A generic vector of the Hilbert space is then uniquely identified is by a net of complex numbers such that
(16) 
Matrix elements of the operators , , and can be easily computed in this representation by using the commutation relations (13) and the condition (14). In particular the number operator
(17) 
is diagonal. Every other Hilbert space representations of the CCR satisfying the hypotheses of the StoneVon Neumann theorem is unitarily equivalent to this one.
2 Infinite Systems. Free fields in Minkowski space
The situation drastically changes when considering systems with infinitely many degrees of freedom: the StoneVon Neumann theorem is no longer applicable and there exist uncountably many inequivalent Hilbert space representations of the canonical commutation relations (4) or (11) when (see e.g. Strocchi (1985); Haag (1996)). The distinction, then, between a CCR algebra and a its Hilbert space representations becomes crucial; finding a quantum theory for an infinite system (such as a field) involves therefore two distinct steps:

construction of an infinite dimensional algebra describing the degrees of freedom of the quantum system;

construction of a Hilbert space representation of that algebra.
The final step would consist in choosing, among the infinitely many inequivalent representations, a physically meaningful one. Unfortunately, a complete classification of the possible representations of the canonical commutation relations does not exist and is not foreseen in the near future. This lack of knowledge is especially relevant in curved backgrounds where, generally speaking, the selection of a representation cannot be guided by simple physical principles as it is the case in flat space. Indeed, while the CCR’s have a purely kinematical content, the construction and/or the choice of one specific representation in a Hilbert space is always related to dynamics and different dynamical behaviors require inequivalent representations of the CCR’s (see e.g. Strocchi (1985); see however the discussion in halver ); in particular the choice of a specific representation is related to many fundamental issues such as renormalizability, thermodynamical equilibrium and entropy, symmetries, phase transitions, etc… The purpose of the following discussion is to clarify some aspects the above nonuniqueness features by a detailed discussion of the most elementary and wellknown case: the KleinGordon field in Minkowski space.
2.1 Canonical quantization of a Minkowski KleinGordon field
2.1.1 Classical field theory
denotes the dimensional Minkowski spacetime; an event is parameterized by a set of inertial coordinates . The nonvanishing covariant components of the metric tensor are in every inertial frame; the Lorentzinvariant product of two events is given by .
The dynamics of a real massive KleinGordon scalar field defined on is governed by the Lagrangian density
(18) 
which implies the KleinGordon equation:
(19) 
At a given time , the canonical variables encoding the degrees of freedom of the field are the values of field itself and of the conjugated momenta
(20) 
Here the spatial coordinates are used as continuous labels for the infinitely many degrees of freedom of the field in very much the same way as the discrete indices of the finite dimensional case label the degrees of freedom of the mechanical system (2). Consequently, the Poisson brackets of the canonical variables and can be formally assigned as in (2):
(21) 
the derivatives with respect to the variables and with continuous indices or in the Poisson brackets are now functional derivatives. The interpretation of the above canonical structure is as follows. Under suitable assumptions, the assignment of initial values at a given time
(22) 
uniquely identifies a solution of the field equation (19). This means that the phase space of the classical KleinGordon field theory can be identified with the space of the possible initial condition at a given time, i.e. a point of the phase space is a pair of suitably smooth functions rapidly decreasing at infinity. The linearity of the field equation (19) implies that the phase space also has a linear structure. The symplectic structure of the phase space is encoded in the symplectic twoform
(23)  
(24) 
that accounts for the formal Poisson brackets defined above; here the integral is overall the space manifold . By using the KleinGordon equation (19) it is readily seen that the symplectic form does not depend on the time coordinate :
(25) 
The phase space can actually be identified with the linear space of real solutions of the field equation (19) for initial condition belonging to a suitably chosen function space; is endowed with the symplectic form .
2.1.2 Quantization: CCR algebras
Following the discussion of the finitedimensional case, in order to quantize the classical theory with infinitely many degrees of freedom outlined in the previous section, we introduce a fixed time CCR algebra by imposing the following canonical commutation relations:
(26) 
we have used here the same letters and as in the previous section also for the canonical quantum variables and set .
A practical way to implement the CCR’s (26) goes as follows: the symplectic form can be extended to a pseudoscalar inner product in the linear space of complex classical solutions of the KleinGordon equation (19):
(27)  
(28) 
One then looks for set of solutions of the KleinGordon equation labeled by the spatial momenta dual to the “spatial labels” ; the set should be complete and orthonormal in the following sense:
(29) 
A real solution of the KleinGordon equation can therefore be written as the following real superposition of the modes :
(30) 
where the elements of the CCR algebra and are related to by
(31) 
The standard plane wave solution to these requirements is written as follows
(32) 
where
(33) 
These waves have “positive frequency”
(34) 
i.e. they are eigenfunctions of the global timelike Killing vector field with positive eigenvalue . One easily checks that the conditions (29) are verified for the modes(32). The field and its canonically conjugated momentum admit therefore as following real mode expansions:
(35)  
(36) 
Now, assume that the coefficients and satisfy the following canonical commutation relations in space:
(37) 
the fundamental space canonical commutation relations (26) immediately follow from (35) and (36).
2.1.3 Representing the quantum field in a Hilbert space I
Up to this point we have constructed the quantum KleinGordon field as a linear combination of the elements of the canonical algebra (37). We have not yet represented the field and the canonical commutation relations by operators in a Hilbert space.
The first observation is that there exists a natural representation
(described at beginning of every book about QFT) which is most obviously related to the mode expansion (35) and which can be constructed starting from the vacuum vector annihilated by the (distributional) operators :
(41) 
The other vectors of the Hilbert space of this natural but by no means unique representation are constructed in perfect analogy with the finite dimensional case. The vacuum vector is the (unique) ground state for the energy operator and contains no particle; the particle states are constructed by repeated application of the creation operators to the vacuum and have positive energy in every Lorentz frame. The twopoint vacuum expectation value of the field (in short: the twopoint function ^{3}^{3}3For free fields a twopoint function characterize completely a corresponding Fock representation. The Fock representation in turn can be constructed by applying the field algebra to a certain reference cyclic vector often called the “vacuum” even when it is not void. All these words (twopoint function, representation, vacuum, state) are used interchangeably ) has the following expression:
(42)  
(43) 
Since we are dealing with a free field, the twopoint function contains all the information about the theory; this point will be discussed later in detail. One can easily verify that the covariant commutator coincides with , see Eq. (38):
(45) 
2.1.4 Representing the quantum field in a Hilbert space II
Any Hilbert space representation of the quantum field must realize the equaltime CCR’s (26); equivalently, the covariant commutator of two field operators, computed in the chosen representation, must coincide with the distribution (38). Consider for instance the following change of basis in the canonical algebra (37):
(46) 
It follows immediately that
(47) 
and the transformation is canonical. Eq. (46) defines a special instance of Bogoliubov transformation. The (algebraic) field
(49)  
by construction is the same field as the one given in (35); it solves the KleinGordon equation and provides an algebraic implementation of the covariant commutator (38). We can however use the expression (49) to build another representation ^{4}^{4}4We use the same symbol to indicate a generic operator in a Hilbert space without reference to any specific representation. ; starting from the “vacuum” annihilated by the operators :
(50) 
and proceeding as before. This representation is fully characterized by the twopoint function
(51)  
(52) 
A straightforward computation shows that the field has the right covariant commutator
(53) 
and the quantization is canonical. When the Bogoliubov transformation (46)is not unitarily implementable this construction gives rise to an inequivalent Hilbert space representations of the CCR’s and of the field . In particular, although the commutator is covariant, the Poincaré symmetry is in general broken. An example is provided by choosing equal to a constant . In this case the twopoint function can be easily expressed in terms of
(54) 
and the spacetime translation symmetry is manifestly broken. We do not dwell here on the mathematical conditions that discriminate between inequivalent representations and refer the reader to the literature. The important point is that inequivalent representations describe inequivalent physical situations (if any physical interpretation is available).
2.1.5 Representing the quantum field in a Hilbert space III
Bogoliubov transformations are not the end of the story. The fundamental space CCR’s (26) that account for the degrees of freedom of the quantum system can be realized in another way by making use of two independent copies of the momentum space CCR algebra (37). Consider indeed a second independent canonical algebra
(55) 
and write
(56) 
It follows immediately that the momentum space CCR’s also hold for the newly introduced symbols: etc.. Note however that the relations (56) cannot be inverted. The algebraic quantum field
(57)  
(58) 
is canonical and, as anticipated, the fundamental space CCR’s (26) have been implemented by using two independent copies of the space CCR’s (37).
The twopoint function in the vacuum annihilated by the operators and has the following expression:
(59) 
A particularly important class of representations in the family of states (59) is obtained by choosing the following functions
(60) 
correspondingly
(61) 
These twopoint functions are of fundamental importance in quantum field theory as they provide the KuboMartinSchwinger (KMS) thermal representations of the KleinGordon field at inverse temperature Birrell and Davies (1982); Kapusta (1989); Bros and Buchholz (1992). Every inverse temperature describes a physically distinct situation and the corresponding representations are inequivalent. The ground state is recovered in the limit . One can appreciate here concretely the fact that inequivalent representations of the same canonical algebra have distinct physical meaning.
The method of doubling the CCR algebra as in Eq. (56) is at the basis of the socalled thermofield theory Umezawa et al. (1982). It may however be worthwhile to insist that there is no doubling of the degrees of freedom in representing one and the same field algebra (26). As we will see below, the doubling of the degrees of freedoms is an artifact of the momentum space representation used to implement the xspace CCR’s.
3 Wightmantype approach
For a free field, knowing the twopoint function gives all the information about the theory. All the examples presented in the previous section can be revisited by taking the twopoint function as the starting point. In this section we will elaborate on this viewpoint in order to prepare the discussion of the general curved case.
An acceptable twopoint function of a KleinGordon field is a positivedefinite^{5}^{5}5The positivedefiniteness hypothesis must however be relaxed when dealing with gauge QFT Strocchi (1985) distribution (or some other distribution space) that solves the KleinGordon equation w.r.t. both variables and :
(62) 
The condition of canonicity is written more conveniently by using the covariant commutator: is required to be a solution of the functional equation
(63) 
The Hilbert space of the theory can be reconstructed by using standard techniques Streater and Wightman (1989); Reed and Simon (1980). The property of positivedefiniteness of the twopoint function is used to defines a natural preHilbert scalar product in the space of test functions Reed and Simon (1980):
(64) 
The one particle Hilbert space is obtained by Hilbert completion of the quotient space where is the subspace of functions having zero norm. The full Hilbert space of the model is the symmetric Fock space . As regards the operatorvalued distribution
(65) 
representing the field, this is expressed in terms of the ladder operators that naturally act in the Fock space (see Eq. (93) below). Any positivedefinite twopoint function solving (62) and (63) therefore provides a canonical quantization of the KleinGordon field.
3.0.1 Spacetime translation symmetry
The additional hypothesis that the spacetime translations be an exact symmetry amounts to requiring that the twopoint function depend only on the difference variable :
(66) 
Eq. (62) is then most easily solved in Fourier space, where it becomes algebraic. The Fourier transform and antitransform of the distribution are introduced as follows:
(67) 
the Fourier representation of the KleinGordon equation (62) and of the functional equation (63) are respectively
(68)  
(69) 
However, translation invariance is not restrictive enough and there are yet infinitely many possibly inequivalent solutions for these equations. They can be parameterized by the choice of an arbitrary function as follows:
(70) 
should be an acceptable multiplier for the distribution so that is a tempered distribution. Taking the Fourier antitransform of we obtain the space representation of the twopoint function:
(71) 
The BochnerSchwartz theorem tells that is positivedefinite if and only if the tempered distribution is a positive measure of polynomial growth. This is in turn guaranteed by requiring that
(72) 
and that the growth of is polynomial. Eq. (70) therefore provides a huge family of translation invariant canonical quantizations of the KleinGordon field. Some physical criterium is needed to discriminate among them: we will discuss a few examples.
3.0.2 Ground state
This representation is selected by imposing the physical requirement of positivity of the spectrum of the energy operator in every Lorentz frame. Equivalently said, the joint spectrum of the energymomentum vector operator must be contained in the closed forward cone. This condition implies that the also the support of must be contained in the closed forward cone Streater and Wightman (1989) and therefore the second term at the RHS of (70) has to vanish i.e.
(73) 
Positivity of the energy spectrum is therefore associated with the most obvious solution of the functional equation (69):
(74) 
The space representation of twopoint function (71) coincides precisely with the expression already given in Eq. (42). The Fourier representation (74) also shows that the theory is invariant also under the restricted Lorentz group.
Finally, by plugging the integral representation (42) into the scalar product (64) we obtain that the oneparticle Hilbert space is concretely realized in momentum space as follows:
(75) 
The relation between the twopoint function and the positive frequency solution of the KleinGordon equation can be further clarified by observing that for any test function , the convolution
(76) 
is a smooth solution of the KleinGordon equation containing only positive frequencies. The pseudoscalar product introduced in (28) is positive on the subspace generated by the positive frequency modes. Hilbert completion of such subspace gives rise to the same oneparticle space.
3.0.3 Other Poincaréinvariant representation
Dropping the requirement of positivity of the energy we immediately find a one parameter family of other possible Lorentz invariant solutions characterized by the choice constant:
(77) 
(78) 
which is a canonical local and Poincaréinvariant solution of the KleinGordon equation. Of course does not satisfy the spectral condition since states with negative energy are now present.
3.0.4 KMS representations
Another split of the commutator is obtained by replacing the Heaviside step function in (70) by the BoseEinstein factor as follows Bros and Buchholz (1992):
(79) 
so that
(80) 
and the condition (69) is manifestly satisfied. Fourier antitransforming brings us back to the twopoint function given in Eq. (59). Furthermore, since is a positive measure and therefore is positive definite.
At this point we step back and observe that the difference between the twopoint functions (51) and (59) is just the last term at the RHS of Eq. (51)
(81) 
which is absent in Eq. (59). Two remarks are in order here:
1) since both theories are canonical the contribution of to the commutator must be zero. This is indeed true because .
2) Adding any symmetric positive function to an acceptable twopoint function produces another acceptable twopoint function (provided positivedefiniteness continue to hold.
4 Extended Canonical Formalism
4.0.1 General Scheme
We now summarize the findings and the hints of the previous section in a general framework suitable to discuss the problem of quantizing a field on a curved spacetime. What follows is an account of an extension of the quantum canonical formalism that we have recently introduced in Moschella and Schaeffer (2008). We limit the discussion to the case of a real KleinGordon field on a Lorentzian manifold .
A quantum field is a distributional map from a space of test functions into the elements of a field algebra :
(87) 
The algebraic structure is given by the commutation relations: the commutator has a purely algebraic or kinematical content and, for free fields, is a multiple of the identity element of the algebra:
(88) 
is an antisymmetric bidistribution on the manifold , solving the KleinGordon equation in each variable and vanishing coherently with the notion of locality inherent to , i.e. for any two events spacelike separated; shortly denotes the invariant volume form.
Quantizing means representing the above commutation relations by an operatorvalued distribution in a Hilbert space i.e.
(89) 
For free fields the truncated point functions vanish and the theory is completed encoded in the knowledge of a positive semidefinite twopoint function , a distribution whose interpretation is the twopoint ”vacuum” expectation value of the field:
(90) 
As discussed in the previous sections, the following is the crucial property that