The counting of Nambu-Goldstone bosons in a non-Hermitian field theory

We demonstrate that the number of Nambu-Goldstone bosons is always equal to the number of (relevant) conserved currents inside the scenario of non-Hermitian field theories with spontaneous symmetry breaking. This eliminates the ambig\"uity which normally appears in Hermitian field theories, specifically in the non-relativistic regime.


Introduction
When the symmetry of a system, represented by a Lagrangian, is spontaneously broken, gapless particles dubbed Nambu-Goldstone bosons appear. The Nambu-Goldstone theorem then suggests that the number of broken symmetries is equivalent to the number of Nambu-Goldstone bosons [1]. Exceptions to this rule appear when a pair of broken symmetries correspond to only one Nambu-Goldstone boson [2,3,4]. Another interesting aspect is that in the standard formulation of the Nambu-Goldstone theorem, the Nambu-Goldstone bosons (being massless particles) are expected to have a linear dispersion relation. However, for the cases where two broken symmetries correspond to a single Nambu-Goldstone boson, it comes out that the dispersion relation is surprisingly quadratic [4]. Nielsen and Chadha formulated an interesting explanation of this phenomena in [2]. Subsequently, Nambu explored the same problem in [5]. Murayama, Watanabe and Brauner formulated some equations explaining the counting rule for the Nambu-Goldstone bosons by extending the arguments of Nielsen and Chadha as well as Nambu's arguments [3]. In [4], the author demonstrated the generic character of this phenomena which is a natural consequence of the fundamental connection between the Nambu-Goldstone theorem and the Quantum Yang-Baxter equations (QYBE). The formulation of the Nambu-Goldstone theorem in the scenario of the non-Hermitian Field theory (respecting the Parity-time reversal (P T ) symmetry in order to satisfy unitarity) was done by Alexandre, Ellis, Millingtone and Seynaeve in [6]. In such a case, the authors demonstrated that for every conserved current, there should be a gapless mode representing the Nambu-Goldstone boson. The conserved current in these situations is not related to a symmetry of the Lagrangian due to the non-Hermitian character of the theory [7]. In this paper, by using the Kähllen-Lehman spectral representation, we demonstrate that in the relativistic limit, the dispersion relation of the Nambu-Goldstone bosons is linear and that the number of independent conserved currents is equal to the number of Nambu-Goldstone bosons. In the non-relativistic limit, we demonstrate that for each pair of conserved currents, one of them becomes negligible in this limit, reducing in this way the number of independent (relevant) conserved currents. The remaining number of independent currents, is equal to the number of observed Nambu-Goldstone bosons, this time with quadratic dispersion relation as it should be in this limit. All these results emerge naturally in the non-Hermitian approach proposed in [6,7]. The paper is organized as follows: In Sec. (2), we review the formulation of the Nambu-Goldstone theorem when the Lagrangian under study is non-Hermitian and satisfying the P T -symmetry. We then obtain the conditions under which the dispersion relation of the Nambu-Goldstone bosons becomes quadratic by using the Källen-Lehman representation. In Sec. (3), we extend the analysis of the Nambu-Goldstone theorem by expressing the order parameter as a double-commutator and then we evaluate the number of independent terms. In Sec. (4), we analyze the number of independent currents for the relativistic and non-relativistic cases by using the Källen-Lehmann representation, demonstrating that the number is reduced by one-half in the non-relativistic limit and it always matches with the number of Nambu-Goldstone bosons. Finally in Sec. (5), we conclude.
2 The Nambu-Goldstone theorem: Non-Hermitian formulation As a starting point, we can assume a transformation for the field Φ in the following form where T is a generator for the transformation. Note that for the case of non-Hermitian Lagrangians we cannot consider symmetry transformations of the Lagrangian, but rather transformations corresponding to a conserved current j µ [6,7]. The corresponding conserved charge is Q = d 3 xj 0 (x). The important Mathematical object to be analyzed is the spontaneous symmetry breaking condition (2) Here the left-hand side is a commutator of the Nambu-Goldstone field and the conserved current under analysis. The right-hand side corresponds to the order parameter. Note that for the case of non-Hermitian field theories, the inner products are defined with respect to the C ′ P T transformation, which is analogous to the standard Charge conjugation-Parity-Time reversal transformation (CPT) defined in the Fock space [6]. Here however, the operator C ′ does not represent charge conjugation. This is just an operator representing the transformation which guarantees a positive definition of the inner product inside the P T symmetric formulation. Then the bar symbol over the vacuum state projected to the left in eq. (2), is an important distinction to consider inside this formulation. In the same way, the complete set of intermediate states will be defined asÎ = |N ><N |. This means that the covariant version of eq. (2), can be expressed as By using the Källen-Lehman spectral representation, we can simplify the previous expression as If we continue the standard procedure, without any surprise, we will obtain a linear dispersion relation. However, if we analyze the previous expressions in detail, we can extract the conditions under which, the dispersion relation becomes quadratic and then we can proceed to analyze the possible consequences of this issue. We will first explore the conditions that the Källen-Lehman spectral representation has to satisfy in order to obtain a quadratic dispersion relation.

Conditions for the dispersion relation to be quadratic based on the Källen-Lehmann representation
Following the same arguments illustrated in [4], we can obtain the general conditions for the dispersion of the Nambu-Goldstone bosons to be quadratic. The Källen-Lehmann representation by itself cannot tell us anything about the counting of Nambu-Goldstone bosons, at least in its standard form. The same representation can however, help us to understand under which circumstances we can expect the quadratic dispersion relation. The starting point is to rewrite the result (4) in a more explicit form as This previous expression can be factorized as Here p µ = (E, p) andp µ = (−E, p). Then in order to analyze the previous expression properly, it is convenient to expand its product with respect to a form n µ = (1, 1) as n µ <0|[j µ (y), Φ(x)]|0 >= 4πiΘ(p 0 )ρ(p 2 )e −ip 0 (y 0 −x 0 ) (Ecos(p n · (y − x)) + ipsin(p n · (y − x))) . (7) Here we have considered that ρ(p 2 ) =ρ(p 2 ) in agreement with the arguments of causality illustrated in [4]. Note that in the non-relativistic case E >> p, then we obtain from the previous expression Since the cosine function expands in a quadratic form in its argument at the lowest order, then we have demonstrated that E ≈ |p| 2 for this case. This can be observed from eq. (8) because the energy in addition expands linearly at the lowest order. The result is consistent with the claims that in non-relativistic systems, the Nambu-Goldstone bosons have a quadratic dispersion relation [1]. Note that our derivation is not concerned with the exact functional dependence of ρ(p 2 ). On the other hand, in the relativistic limit, we have E ≈ |p|, and then by multiplying the result (5) with the form n µ , we get For convenience and clarity, we have separated the temporal part and the spatial part in the exponential term. This expression clearly shows that the energy and the momentum go simultaneously to zero linearly when the system is relativistic. This means that E ≈ |p| for the massless Nambu-Goldstone boson as it is expected in this limit. Note that here nothing can be concluded about the relation between the number of Nambu-Goldstone bosons and the conserved currents. Something important to remark is that in order to understand the way how the Nambu-Goldstone disperse, we must focus on the behavior of the phases e ±ipy and the way how they factorize at the end. This was the key part in the analysis done in [4].

Double-commutator structure: The standard case based on broken generators
In order to find explicitly the connection between the number of Nambu-Goldstone bosons and the number of independent currents, we have to develop the expression (5) further. Before doing this, in the ordinary case, it was assumed that the broken symmetries of the system obeyed the Lie-algebra structure. This suggests that the following expression should be satisfied with C plc denoting the structure constant. By extending the arguments to the case of conserved currents, we can convert the result (7) into the following expression which is a double-commutator with four terms after expanding the expression. If we evaluate explicitly the double commutator, then we obtain We can introduce now a pair of complete set of intermediate states, which for the present non-Hermitian field theory are equivalent toÎ = m,m |m ><m| andÎ = n,n |n ><n|. Then we obtain m,m n,n Here the sum is applied to all the terms. Under the space-time translational invariance assumption, then we get the more explicit expression Previous arguments suggests that the first and the fourth terms of the previous expression represent the same sequence of events [4] after summing over the degenerate vacuum. The same applies for the second and the third term in the previous equation. Such equality corresponds to two pairs of Quantum Yang-Baxter Equations (QYBE). Although the formulation showed here is valid in the ordinary sense, it requires some modifications when we deal with non-Hermitian formulations. In what follows, we will apply the Källen-Lehmann spectral representation. In such a case, since the Wick Theorem forbids non-trivial triple products at the moment of evaluating propagators [8], we have to formulate an equivalent way to represent the triple products appearing previously in the language of the Källen-Lehmann representation. This implies the analysis of the product of a pair of two-point functions or equivalently, exploring the product of two propagators. Note that in this section we developed the standard method which appear in any text-book, nothing different to that. The coming section comes with the new ingredietns to be analyzed.
If we introduce these matrix definitions inside eq. (17), we can then analyze the two regimes, namely, the relativistic regime and the non-relativistic one. Note that the two columns appearing in the previous matrices represent the number of independent conserved currents under analysis. This number is equivalent to the number of Nambu-Goldstone bosons appearing in the system.

Non-relativistic regime
For the non-relativistic regime, E >> p, then the matrices defined in eq. (18), are reduced to only one independent column. In other words, in this regime, all the matrices defined in eq. (18) are reduced to If we introduce this result in eq. (17), then we get Then what was initially a 2 × 2 matrix becomes a single component. This means that in the non-relativistic regime, the number of non-redundant conserved currents is reduced by one half. This in addition means that the number of Nambu-Goldstone bosons is also reduced correspondingly. Before we have demonstrated in eq. (8) that the dispersion relation of the Nambu-Goldstone bosons is quadratic in these cases. Since the the pair of independent currents are reduced to a single one in this case, then we can safely assume that p 0 ≈p 0 and p n ≈p m in eq. (20).
The fact that the matrix preserves the two linearly independent vectors (columns), means that the two currents under analysis, are relevant in this regime. Then the number of Nambu-Goldstone bosons is not reduced in these situations because in general E =Ẽ as well as p n =p m . Already in eq. (9) we have demonstrated that in this regime the dispersion relation is linear in these situations.

Conclusions
In this paper we have demonstrated that the non-Hermitian formulation of Quantum-Field theory avoids the ambigüities in the counting of Nambu-Goldstone bosons by connecting the number of NG bosons directly with the number of independent conserved currents in the system. Then while in the relativistic case we have N Nambu-Goldstone bosons with linear dispersion relation, connected to the existence of N independent conserved currents; in the non-relativistic case, the number of independent conserved currents as well as the number of Nambu-Goldstone bosons is reduced by a half to N/2. The dispersion relation of the Nambu-Goldstone bosons is quadratic as expected in the non-relativistic case. We have used the Kähllen-Lehmann spectral representation for our analysis by expanding the arguments illustrated in [6,7]. Note that for the case of Hermitian theories, the previous results are the same but in such situations, the relation between the number of Nambu-Goldstone bosons and the number of broken symmetries is not fixed when we analyze the different regimes (relativistic or non-relativistic). This means that while the standard Hermitian formulation of Quantum Field Theory might shows some ambigüity in the counting rules for Nambu-Goldstone bosons, these ambigüities disappear for the case of non-Hermitian field theories. These important aspects remark the importance of understanding in deep detail the non-Hermitian formulations as those proposed in [6,7].