Pseudo-Hermitian approach to Goldstone’s theorem in non-Abelian non-Hermitian quantum field theories

Abstract: We generalise previous studies on the extension of Goldstone’s theorem from Hermitian to non-Hermitian quantum field theories with Abelian symmetries to theories possessing a glocal non-Abelian symmetry. We present a detailed analysis for a nonHermitian field theory with two complex two component scalar fields possessing a SU(2)symmetry and indicate how our finding extend to the general case. In the PT-symmetric regime and at the standard exceptional point the Goldstone theorem is shown to apply, although different identification procedures need to be employed. At the zero exceptional points the Goldstone boson can not be identified. Comparing our approach, based on the pseudo-Hermiticity of the model, to an alternative approach that utilises surface terms to achieve compatibility for the non-Hermitian system, we find that the explicit forms of the Goldstone boson fields are different.


Introduction
The extension from quantum field theories with Hermitian actions to those with a non-Hermitian actions has been addressed recently for various concrete systems, such as scalar field theory with imaginary cubic self-interaction terms [1,2], field theoretical analogues to the deformed harmonic oscillator [3], non-Hermitian versions with a field theoretic Yukawa interaction [4,5,6,7], free fermion theory with a γ 5 -mass term and the massive Thirring model [8], PT -symmetric versions of quantum electrodynamics [9,10] and PT -symmetric quantum field theories in higher dimensions [11].
The generalisations also include Goldstone's theorem [12,13] and the Higgs mechanism [14,15,16] [17,18,19,20]. Both of these mechanisms are governed by the continuous symmetries of the theories, global or local, respectively, that might by spontaneously broken by some vacuum states. The special feature of non-Hermitian systems is that an additional discrete antilinear symmetry [21] is superimposed on top of the continuous symmetries, that can also be spontaneously broken, albeit not exclusively for the ground state in this case. The regime in which the discrete symmetry is broken is regarded as unphysical. In general, the antilinear symmetry separates the parameter space of the theory into regimes of different types of behaviour. The physical subspace is bounded by the values for which the eigenvalues of the mass squared matrix acquire an exceptional point, a singularity or become zero. It is the interplay between these two types of symmetries, continuous and discrete, that produce very interesting and novel behaviour when compared to the standard Hermitian setting.
There is a well known problem that seems to suggest that non-Hermitian quantum field theories are inconsistent, see e.g. [22,23,8]. However, just as for non-Hermitian quantum theories [24,25,26,27] there are methods and techniques to overcome these issues to obtain a perfectly consistent theory. The conundrum for the quantum fields theories consists of the feature that the two sets of equations of motion, derived from functionally varying the action with respect to the scalar fields on one hand and with respect to their complex conjugates on the other, are incompatible. So far two distinct alternative propositions have been made to overcome this issue. Alexandre, Ellis, Millington and Seynaeve proposed to apply a non-standard variational principle by keeping some non-vanishing surface terms [17,20] or, in line with the pseudo-Hermitian/PT -symmetric quantum mechanical approach [24,25,26,27], one may seek a consistent equivalent similarity transformed Hermitian action, as pursued by Mannheim and the present authors [18,19]. While some features are the same in both approaches, e.g. both versions predict the same number of massless Goldstone bosons that is expected from Goldstone's theorem, they also differ in several aspects. While in the former proposition Noether's theorem is evaded the latter is based on the standard variational principle leading to standard Noether currents. Moreover with regard to the Higgs mechanism the "surface term approach" predicts that the gauge particle becomes massive in the local case [20], whereas the "pseudo-Hermitian approach" leads to a theory in which the gauge particle remains massless at the exceptional point [18]. Here we also find that the explicit form of the Goldstone bosons differs.
Previous considerations were focused on the analysis of non-Hermitian systems with a global and local Abelian U (1)-symmetry, they were recently extended to non-Abelian theories within the surface term approach [20]. Here we also extend these studies to the non-Abelian case by applying the pseudo-Hermitian approach. We analyse in detail a non-Hermitian scalar field theory with two complex two component scalar fields possessing a SU (2)-symmetry and an overall discrete antilinear symmetry. We compare our results to those obtained in [20] by means of the surface term approach.
In section 2 we discuss the generalities of the pseudo-Hermitian approach to achieve compatibility in non-Hermitian quantum field theories, with an emphasis on how it modifies the identification of the mass squared matrix and Goldstone's theorem. In section 3 we discuss a concrete model with two complex scalar fields in the fundamental representation, by deriving an equivalent Hermitian action for the model, discussing its SU (2)-symmetry, its vacua, mass squared matrices, physical regions and identifications of the Goldstone bosons in the different regimes. We state our conclusions in section 4.

Pseudo-Hermitian approach to spontaneously broken symmetries
We consider here complex scalar quantum field theories described by actions of the following generic type with n-component complex scalar fields φ = (φ 1 , . . . , φ n ) and potential V (φ). The action is assumed to possess three general properties: i) It is invariant under a global continuous symmetry φ → φ+δφ with V (φ) = V (φ+δφ). The symmetry is, for instance, generated by a Lie group g with Lie algebraic generators T , so that being global implies an infinitessimal change δφ = αT φ with α being a small parameter and ∂ µ (αT ) = 0. ii) It is invariant under a discrete antilinear symmetry φ(x µ ) → U φ * (−x µ ), with U being a constant unitary matrix. These symmetries may be viewed as modified CPT -symmetries. When U → I the symmetry reduces to the standard CPT -symmetry. iii) The potential V (φ) is not At first sight such type of theories appear to be inconsistent as the two sets of equations of motion obtained by functionally varying the action I separately with respect to the fields φ i and φ * i , δI n /δφ i = 0 and δI n /δφ * i = 0, are in general incompatible when U = I. One may, however, overcome this problem by using a non-standard variational principle combined with keeping some non-vanishing surface terms [17,20] or alternatively by exploiting the fact that the content of the theory is unaltered as long as the equal time commutation relations are preserved and carry out a similarity transformation that guarantees that feature [8,18,19]. Hence, in the latter approach one seeks a Dyson map η, named this way in analogy to its quantum mechanical counterpart [28], to construct a new equivalent actionÎ with the difference that now the transformed potential is HermitianV (φ) =V † (φ). The matrixÎ is a result of the similarity transformation. Next it is in general useful to convert the complex scalar field theory into one involving only real valued fields by decomposing the n complex scalar fields into real and imaginary parts as φ = 1/ √ 2(ϕ + iχ) with ϕ,χ ∈ R. Defining then a real 2n-component field Φ = (ϕ 1 , . . . , ϕ n , χ 1 , . . . , χ n ), possibly with the fields in different order to block diagonalize the mass squared matrix, the new actionÎ may be re-written aŝ Analyzing the action in this form, the extension of Goldstone's theorem from the Hermitian to the non-Hermitian case is easily established. At first we identify various types of vacua Φ 0 by solving The continuous global symmetry Φ → Φ+δΦ, i.e.V (Φ) =V (Φ+δΦ) =V (Φ)+∇V (Φ) T δΦ, then implies Differentiating this equation with respect to Φ j and evaluating the result at a vacuum Φ 0 , determined by (2.4), yields Since the last term vanishes, due to (2.4), we are left with two options to solve (2.6).
Either the vacuum is left invariant such that δΦ i (Φ 0 ) = 0 or the vacuum breaks the global symmetry and δΦ i (Φ 0 ) = 0. Denoting θ 0 := δΦ i (Φ 0 ) and multiplying (2.6) byÎ we obtain where H(Φ 0 ) is the Hessian of the potentialV (Φ) evaluated at the vacuum Φ 0 and M 2 is the mass squared matrix. The occurrence of the matrixÎ results from the similarity transformation and is therefore the trace of the feature that the potential is non-Hermitian. It also has the effect that M 2 is no longer Hermitian either. We can now read off Goldstone's theorem for non-Hermitian systems from (2.7). When the vacuum is left invariant by the global symmetry transformation we have θ 0 = 0 so that there is no restriction on M 2 . However, when the vacuum breaks the global symmetry we have θ 0 = 0 so that θ 0 becomes a null vector for M 2 . Thus, in this case we have a zero mass particle, that is identified as a Goldstone boson.
Assuming that the symmetry is generated by a Lie group g, we may repeat this argument for each Lie algebraic generator T so that we obtain a Goldstone boson for each generator that when acting on the vacuum Φ 0 produces a different one. The crucial difference, when compared to the scenario with Hermitian potentials, is that here M 2 is also not Hermitian. This means that the physical regimes are determined by the discrete antilinear symmetries. Referring to this symmetry as PT -symmetry [25,27] in a wider sense, we may encounter PT -symmetric regimes with real mass spectra, exceptional points with non-diagonalisable mass matrix, zero exceptional points, singularities and a spontaneously broken PT -symmetric regime with unphysical complex conjugate masses. As shown in [19] the identification of the Goldstone boson is different in these regimes and in parts impossible.
Below we will also make use of the general property that the expansions around two vacua, say φ 1 0 and φ 2 0 , that are related by the symmetry transformation T of the potential to theories with mass squared matrix possessing the same eigenvalues. This can be seen from As the kinetic term is invariant by itself no modification of the mass squared matrix will arise from there, apart form the multiplication byÎ as a result of the non-Hermitian nature. Thus we may employ the symmetry to transform the vacuum into the most convenient form for analysis without altering the physics, such as the eigenvalue spectrum of the mass matrix.

A CPT -symmetric non-Hermitian model with global SU(2)-symmetry
Let us now verify the previous general statements for a more concrete system. We consider the action where the two complex scalar fields , are taken to be in the fundamental or spin 1/2 representation of SU (2) and g, µ ∈ R are constants. We allow here for m i ∈ R or m i ∈ iR, so that m i → c i m i with c i = 1 or c i = −1, respectively, takes care of these two possibilities. For simplicity we suppress the parameters c i until we analyse the physical parameter space in section 3.5. We observe that the action I su2 has the aforementioned three properties. It is invariant under a global continuous symmetry , with σ 3 denoting one of the Pauli spin matrices, and the potential V (φ) in (3.1) is evidently not Hermitian. We note that in the surface term approach [20] the antilinear symmetries are implemented differently by PT not acting on the arguments of the fields.

Equivalent Hermitian actions
More explicitly in components and transformed to the real fields ϕ k j , χ k j ∈ R, via φ k j = 1/ √ 2(ϕ k j + iχ k j ), the action I su2 reads As indicated above, the direct functional variation of this action will lead to inconsistent equations of motion and we therefore seek a suitable similarity transformation to resolve this issue. Using the Dyson map , the adjoint actions of η on the real and complex scalar fields and canonical momenta is computed to Thus we can utilize η to transform I su2 into a Hermitian action It is useful to note here for our analysis and especially with regard to the generalizations to systems with symmetries of higher rank that the actionÎ su2 can also be cast into a more compact form aŝ where we defined the matrices and vectors

SU (2) and CPT ± -symmetry
Let us now analyze the modelÎ su2 in more detail. First we verify the SU (2)-symmetry of the action and its effect on the different types of fields. Noting that the change in the complex scalar fields is δφ k j = iα a T kl a φ l j , with the generators T a of the symmetry transformation taken to be standard Pauli matrices σ a , a = 1, 2, 3, we directly identify the infinitessimal changes for the real component fields as It is easily verified that the Hermitian actionÎ su2 remains invariant under the transformations (3.10), (3.11). For the 4 and 8-component fields the symmetries (3.10), (3.11) then translate into with ⊗ denoting the standard tensor product. These expressions may be applied to the action in the forms (3.7) and (3.8), respectively, to verify the SU (2)-symmetry.

SU (2)-symmetry invariant and breaking vacua
Let us now compute the vacua from (2.4) with potential as specified in (3.6). We find there are only two types of vacua, that either break or respect the SU (2)-symmetry,  respectively. We introduced the notation x := ϕ 0,1 1 , y := ϕ 0,2 1 , z := χ 0,1 1 , for the vacuum field components and a := µ 2 /m 2 2 , R := r 2 − (x 2 + y 2 + z 2 ), r := 4 µ 2 + m 2 1 m 2 2 /gm 2 2 for convenience. We note that the defining relation for R can be interpreted as a three sphere in R 4 with center (0, 0, 0, 0) and radius r, which is the geometrical configuration expected from its topological isomorphism with the SU (2)-group manifold. We note that the points µ 2 = −m 2 1 m 2 2 are special as there the three sphere collapses to a point and the symmetry of the vacuum is restored F b 0 → F s 0 . The symmetry properties of the vacua are easily established. Identifying the generators T a of the symmetry transformation as Pauli matrices, where we drop the usual factor of 1/2, we compute the action on the vacuum states, say φ 0 j = (φ 0,1 j , φ 0,2 j ) T for j = 1, 2. We find so that for non-zero fields the vacuum will always break the symmetry with respect to the action of T 1 and T 2 . The action of T 3 seems to require only φ 0,2 j = 0, in order to achieve invariance. However, apart from F s 0 there is no possible choice for the fields in F b 0 so that φ 0,1 j = 0 in that case. Let us now make use of the argument in (2.8) and employ the SU (2)-symmetry to transform the vacuum F b 0 into a physically equivalent, but more manageable one. Choosing two simple target vacuaφ by using the well known formula e iρn·σ = cos ρI + i cos ρ(n · σ) with ρ = α 2 1 + α 2 2 + α 2 3 , n = (α 1 , α 2 , α 3 )/ρ and T a = σ a . The vacuum fields are parametrized as (3.23) so that the form of the target vacuum is motivated by setting x = y = z = 0. We only keep one of the sign in (3.18) and solve (3.21), (3.22) by so that R = r cos ρ. For the vacuum F b 0 this translates with (3.14) into We note that det T = 1 and as required T T = T −1 . EvidentlyF b 0 is of a more convenient form of the vacuum than F b 0 and we shall therefore use it from here on.

Mass squared matrices and null vectors
Next we use the different vacua and expand the potentials around them to determine the mass squared matrix according to the definition in (2.7). Expanding first around the SU (2)-symmetric vacuum F s 0 we find the mass squared matrix with two fourfold degenerate eigenvalues Expanding instead around the SU (2)-symmetry breaking vacuum F b 0 , we obtain the mass squared matrix (3.30) The expansion aroundF b 0 yields the same matrix with ϕ 1 1 = χ 1 1 = ϕ 2 1 = 0. As expected from (2.8) and (3.25), both matrices share the same field independent eigenvalues, that is two different ones each with a threefold degeneracy and two eigenvalues that may give rise to an exceptional point For convenience we defined here K := 3µ 4 /2m 2 2 + m 2 1 − m 2 2 /2 and L := µ 4 + m 2 1 m 2 2 . We confirm the expectation from Goldstone's theorem to find three massless Goldstone bosons in the symmetry breaking sector, since none of the three SU (2)-generators leaves the vacuum F b 0 invariant. According to the relation (2.7) we may compute the corresponding null vectors directly from the SU (2)-symmetry transformation. When applying the infinitessimal changes for the component fields

Physical regions
We will now analyse the parameter space of the system and identify the physical regions based on a meaningful mass squared matrix. To cover all possible cases we are setting therefore in all expressions m 2 i → c i m 2 i .For the model expanded around the broken vacuum the physical regions are then determined by λ b ± ≥ 0, λ b 4,5,6 ≥ 0 corresponding to the four inequalities for the four cases c 1 = ±1, c 2 = ±1. All constraints can be expressed as functions of the two ratios (µ 4 /m 4 1 , m 2 2 /m 2 1 ). We find that no solutions exists for c 1 = c 2 , apart from setting µ = m 2 = 0, so that in these two case the model is unphysical. The physical regions for the remaining two cases c 1 = − c 2 = ±1 are depicted in figure 1. The two different cases depicted in figure 1 do not have any physical regions that intersect. The case c 1 = − c 2 = 1 was also analysed within the surface term approach in [20] and our results appear to match exactly. The case c 1 = − c 2 = −1 was not dealt with in [20], but as depicted in figure 1, it also contains a well defined small physical region. We note that for our model with two complex scalar fields the physical regions have no boundary corresponding to singularities, which appears to be a feature only occurring for the theories with more complex scalar fields, see [19].
Finally in figure 2 we also depict the physical regions for the model expanded around the SU (2)-invariant vacuum.
Here only the case c 1 = c 2 = 1 does not contain a physical region apart from µ = m 2 = 0. The three different cases depicted in figure 2 do not have any physical regions that intersect, apart from the small region near the origin. Comparing figures 1 and 2 we also notice that cases with equal choices for the c i do not share physical regions. This implies that for any particular physical model the breaking of the SU (2)-symmetry leads to an unphysical model and in reverse also that some unphysical models become physical when the SU (2)-symmetry is broken.

The Goldstone bosons in the PT -symmetric regime
We may now compute the Goldstone bosons in terms of the original fields in a similar fashion as discussed in [19]. Defining for this purpose the remaining right eigenvectors v i , i = 4, . . . , 8, and a matrix U containing all of them as column vectors as we diagonalize the mass squared matrix by means of the similarity transformation . For µ 4 = m 4 2 and K 2 = −2L, that are the zero and standard exceptional points, we define the fields ψ i with masses m i by re-writing the squared mass term as Hence, the three Goldstone fields are identified as Setting in M 2 b the fields χ 0,1 1 , ϕ 0,1 1 , ϕ 0,2 1 to zero we compute with det U = 2µ 2 (µ 4 − m 4 2 ) 3 √ K 2 + 2L, so that the explicit form of the Goldstone boson fields in the original fields result to As U is not invertible at the exceptional points for µ 4 = m 4 2 and K 2 = −2L, we need to treat these cases separately. We note that these expressions differ from those obtained in [20].

The Goldstone bosons at the exceptional point
At the standard exceptional point, i.e. when K 2 = −2L and hence λ b + = λ b − , the two eigenvectors v − and v + coalesce so that the matrix U is no longer invertible and the Goldstone boson fields may take on a different form as found in [19]. Instead of diagonalising the mass squared matrix we can convert it into Jordan normal form by means of a similarity transformation. Making m 1 the dependent variable, the exceptional point occurs when m 2 1 = ±µ 2 − m 2 2 /2 − 3µ 4 /2m 2 2 so that the Jordan normal form becomes We compute now det U = (α − β)(µ 4 − m 4 2 ) 3 . Defining the Goldstone boson fields by the same formal expression as in (3.38), but with U replaced by U e , we obtain at the exceptional point the same expressions as in (3.38). It is worth noting that the two degenerate fields take on the form ψ +,e = ψ −,e = (ϕ 2 2 − χ 2 1 )(αϕ 2 2 − βχ 2 1 ) √ β − α . (3.43) We note that it is by far not obvious that the Goldstone boson fields acquire the same form in the PT -symmetric regime as at the exceptional point. This is more a coincidence due to the special nature of the mass matrix rather than a general feature. When considering models with more than two scalar fields this no longer holds even for the Abelian case as observed in [19]. In [20] this regime was not analysed separately.

Conclusions and outlook
Using a pseudo-Hermitian approach to treat non-Hermitian quantum field theories we found that the Goldstone theorem also holds when the global symmetry group is non-Abelian. The explicit form for the Goldstone boson in the PT -symmetric regime and at the standard exceptional points can be found explicitly, although using different diagonalisation procedures for the mass squared matrix. At the zero exceptional point the Goldstone boson can not be identified. When the analysis of our model overlaps with the one carried out in [20] employing the surface term approach, the physical regions coincide exactly. However, the explicit forms of the Goldstone bosons are different.
There are some obvious further extensions to these investigation, that would be interesting to carry out, such as the treatment of models with different Lie symmetry groups and the augmentation of the amount of complex scalar fields. Most interesting, with regard to the comparison with the surface term approach, is the investigation of the Higgs mechanism within the presented framework as that aspect will produce more features and predictions that are clearly distinct in the two approaches [29].
Furthermore, it would be very interesting to establish a closer link between studies carried on non-Hermitian systems in 1+1 dimensions. In principle, the Goldstone theorem does not apply for dimension d ≤ 2 as in those settings the breaking of continuous symmetries inevitably leads to infrared divergent correlation functions. However, in [30] it was argued that the Mermin-Wagner theorem no longer applies for the continuous SO(N )symmetry with N < 2 as it cannot be realized as unitary operations on a vector fields. This feature was exploited in [30] to identify a Goldstone phase for a non-Hermitian system..