The Metric Operator and the Functional Integral Formulation of Pseudo-Hermitian Quantum Mechanics

Pseudo-Hermitian quantum theories are those in which the Hamiltonian H satisfies H H † (cid:2) (cid:3) (cid:3) (cid:3) 1 , where (cid:3) (cid:6) (cid:3) e Q is a positive-definite Hermitian operator, rather than the usual H H † (cid:2) . In the operator formulation of such theories the standard Hilbert-space metric must be modified by the inclusion of (cid:2) in order to ensure their probabilistic interpretation. With possible generalizations to quantum field theory in mind, it is important to ask how the functional integral formalism for pseudo-Hermitian theories differs from that of standard theories. It turns out that here Q plays quite a different role, serving primarily to implement a canonical transformation of the variables. It does not appear explicitly in the expression for the vacuum generating functional. Instead, the relation to the Hermitian theory is encoded via the dependence of Z on the external source j(t). These points are illustrated and amplified in various versions of the Swanson model, a non-Hermitian transform of the simple harmonic oscillator.

has a completely real spectrum for N ³ 2. They attributed this property to an unbroken PT-symmetry, whereby A rigorous proof [2] of the reality came a few years later by exploiting the ODE-IM correspondence, i.e. the correspondence between ordinary differential equations in their different Stokes sectors and integrable models.
In such cases there exists a similarity transformation from the non-Hermitian H to a Hermitian h: Here r is a positive-definite Hermitian operator (re)introduced by Mostafazadeh [3].
It is related to the Q operator [4], which provides a positive-definite metric for the quantum mechanics governed by H, according to ( 4 ) It is also useful to introduce its square h r º = -2 e Q (5) From Eq.
This replaces the usual Hermiticity requirement on the Hamiltonian. H is said to be quasi-Hermitian [5], or pseudo--Hermitian (1) , with respect to h.
The operator h ºe Q is in fact precisely the metric operator occurring in y j y h j , , because the similarity transformation ¢ = Here r r r h † = = 2 , rather than 1, as would be the case if r were unitary rather than Hermitian, so If the operator ¢ A is Hermitian then A is pseudo-Hermitian: A A † = h h 1 , and is an observable, with real eigenvalues (the same as those of ¢ A ).

Functional integral formalism of quantum mechanics
In the functional integral formulation of standard Hermitian quantum mechanics, the basic object of interest is the vacuum generating functional from which Green functions can be obtained by functional differentiation with respect to j(t).
The fundamental question we are asking here is, what is the corresponding expression in pseudo-Hermitian quantum mechanics? One might perhaps expect something like The method we will use to find the correct expression for Z is to use the similarity transformation between our pseudo-Hermitian theory and the equivalent Hermitian theory, where we know how Z should be written. In this paper we will limit ourselves to a particular soluble model, the Swanson model, which can be formulated as a viable quantum theory in a variety of ways (in fact there is a one-parameter family [6] of hs), of which we will pick the three simplest. In [7] we treated the two cases Q Q x = ( ) and Q Q p = ( ), and in addition the pseudo-Hermitian "wrong-sign" quartic oscillator, i.e. Eq. (1) for N = 4.

Z for various versions of the Swanson model
The Hamiltonian for this model, first introduced in [8 where a and a † are standard lowering and raising operators for a simple harmonic oscillator with unit frequency, and w, a and b are real parameters. H is non-Hermitian for a b ¹ . In terms of x and p, where a = + +

Q Q x = ( )
H can be written as This is a simple harmonic oscillator with frequency W = 2ã b.
which can be achieved by Note that Eq. (13) represents a (complex) canonical transformation between the pairs (x, p) and (X, P). Classically Q appears as the active part of the generator of this canonical transformation, according to which It is also worth noting that to construct the classical Lagrangian corresponding to Eq. (11) we have which differs from a normal (scaled) Lagrangian for the harmonic oscillator with frequency W only by the total derivative Our approach will be to start with the naive form for the Euclidean Z[0] corresponding to H, verify that this is correct by transforming to its Hermitian equivalent, then insert the external source j(t) coupled to the Hermitian observable, and finally transform back to obtain the form of Z[j] for the non-Hermitian Hamiltonian H. In this spirit we suppose that in which we have H written in terms of j and p.
We then complete the square in exactly the same way as in Eq. (12), to obtain This is the only place that Q makes its appearance in this procedure.
which can be achieved by The corresponding classical generating function is F x P xP iQ P We now mimic this procedure in the functional integral, starting again with and completing the square in the manner of Eq. (21) rather than (12). Then Now we restore j, coupled to the observable F in this Hermitian version and work backwards: Rewriting this in terms of the original field j p = - Again Q does not appear explicitly, but now the source j appears in an unfamiliar way, with terms in j& j and j 2 .
As a check of these results let us calculate the expectation values F and F F 1 2 from the expression (30). The first is rather trivial: as expected. However, the second check is more interesting: which is indeed the result to be expected from Eq. (27).

Q Q x p
This was in fact the original similarity transformation found by Geyer et al. [9], according to which Q x p = -+ m( )  x Q x P x We have again checked that we correctly obtain F and F F  Finally we add -jF or -jP to l and try to work backwards.
In this paper we have only used the first option in Eq. (39), thinking of quantum fields rather than their conjugate momenta as the relevant objects. However, we could have used the second option in Section 2. The successful construction of the equivalent Hermitian theory to that with a -gz 4 potential raises hopes that a similar construction, within the functional integral framework, might be possible for the corresponding -gj 4 field theory. Some tentative steps were made in this direction in [11], but the generalization seems far from straightforward. (1) In this context, where h is a positive-definite operator, the first term may be preferable. The PT invariance of the original class of Hamiltonians (1) can be expressed as pseudo-Hermiticity with respect to the indefinite operator P.