1. The Framework of AQFT and Its Significance for the Higgs Mechanism

For Earman (e.g., 2004a, 2004b), AQFT qualifies as a starting point for an analysis of the Higgs mechanism because it illuminates the structural features of SSB. Its basic structure is the net of local observables , with being a diamond (a bounded open region that arises from the intersection of light cones) in—flat or constantly curved—space-time, while states ω are positive linear functionals on the algebra of observables . Assuming the Haag-Kastler axioms, one can prove that each state canonically induces a GNS (Gelfand-Naimark-Segal) representation of this algebra in terms of operators on a Hilbert space and that the algebraic structure and the vacuum state suffice to recover the physical content of the theory. In this setting, a symmetry corresponds to an automorphism of the algebra, that is, a structure-preserving mapping of the algebra onto itself. A symmetry is now said to be spontaneously broken if it cannot be implemented in the vacuum representation by means of a unitary operator.Footnote ¹ While SSB is impossible in finite dimensional quantum mechanics, the emergence of unitarily inequivalent representations is typical for infinite quantum systems, among them ferromagnets where they correspond to directions of global magnetization, and in QFT.

In order to avoid the domain questions plaguing unbounded operators, the C∗ algebraic approach works with the Weyl algebras. For instance, operators describing an n-particle quantum mechanical system have the form and , , such that the canonical commutation relations read , for and . It is possible to combine different observables into a single Weyl operator . If in the GNS representation is strongly continuous in both variables, differentiation yields the familiar operators. While in quantum mechanics strong continuity is only violated for rather pathological cases, it arises in quite a few infinite dimensional systems. It thus makes sense to distinguish a regular representation π of the Weyl algebra over a vector space V if is strongly continuous for all . A state is called regular if its associated GNS construction leads to a regular representation. As we shall see below, this condition fails for Higgs states and Dirac states implementing constraints in quantum theory.

While AQFT provides a precise understanding of the phenomenon of SSB and allows a mathematically rigorous analysis of Goldstone's theorem (see sec. 3), it falls short of integrating the SM and the Higgs mechanism. Moreover, attempts to construct nontrivial models of interacting field theories have been unsuccessful. Within the physics community this has often been counted against AQFT, even more so since at least parts of the conventional methods of QFT, most prominently the renormalization group, have been successfully subjected to mathematical scrutiny. While quite a few philosophers of science have been attracted by the algebraic approach (e.g., Halvorson Reference Halvorson, Butterfeld and Earman2006; Fraser Reference Fraser2011), David Wallace (Reference Wallace2011) has recently robed the above-mentioned complaints of physicists in a philosophical argument. As tempting as it may be, I will not enter into this general discussion here, for the analysis of this article does not require a broad commitment to AQFT, as long as there are no convincing arguments showing that my theses were an artifact of AQFT or a proven no-go theorem barring its applicability to the Higgs mechanism. In my view, AQFT and the axiomatic method in general are neither automatically committed to nor exclusively motivated by foundational aspirations but amount to an essentially critical attitude in the examination of theoretical frameworks—not unlike philosophers’ analyses—and contain an opportunistic potential in experimenting with such frameworks in a mathematically controllable fashion. To my mind, this understanding is largely in agreement with the history of the axiomatic method since the days of David Hilbert and John von Neumann (e.g., Stöltzner Reference Stöltzner, Rédei and Stöltzner2001) and corresponds to how most contemporary investigators in AQFT understand their project.

Detlev Buchholz (Reference Buchholz2008), whose work is perhaps closest in spirit to the original Haag-Kastler program, has emphasized the adaptability of AQFT and its value for the inquiry of the theoretical framework of QFT. Among the concrete advantages of AQFT are that unobservable events play no role for the physical content—it is thus automatically objective in Earman's sense—and that the vacuum has “stronger stability properties than is normally assumed” (48). A basic assumption of AQFT is the primacy of quantum physics over quantization. More concretely, despite its impressive successes as a recipe, “the hypothesis that the strong and weak interactions come about by quantizing a fictive classical field theory seems unnatural and certainly not the last word. Tangibly classical aspects … should rather emerge from the more basic quantum theory” (46). This approach to the relationship between classical and quantum physics can count on some interesting findings. For instance, “there is a series of classical self-interacting field theories to which there exists no corresponding quantized version. One suspects that also the Higgs-model and quantum electrodynamics belong to this family, but there are still gaps in the arguments” (48). It is of course true that even if such an argument were established, we could still change our idea about quantization and adopt the Feynman path integral approach—which has its own mathematical gaps.

2. Earman's Program: Constrained Quantization

Let me now turn to Earman's program to reach an objective, that is, gauge-invariant account of the Higgs mechanism in terms of constrained Hamiltonian dynamics. “When the shift is made from the Lagrangian to the Hamiltonian formulation, constraints will appear; find all of the constraints and single out the first class constraints; quotient out the gauge orbits generated by the first class constraints to get the reduced Hamiltonian phase space whose phase functions are gauge-invariant magnitudes; finally, quantize the unconstrained system to get a quantum field theoretic description stripped of surplus gauge structure” (Earman Reference John2004b, 1239). This approach is motivated by Dirac's (Reference Dirac1964) famous conjecture that identifies first-class constraints as generators of gauge transformations. In Dirac's theory, the constraints emerging in the transition from the Lagrangian to the Hamiltonian define a submanifold of primary constraints (on the cotangent bundle). In a second step, one has to secure the compatibility of these constraints with the Hamiltonian dynamics by demanding for all m, where the weak equality ‘≈’ means equality on the hypersurface defined by these constraints. This yields secondary constraints, and the reduction algorithm might continue to higher orders as long as degrees of freedom remain unfixed.

From the Poisson bracket structure another classification emerges. A constraint ϕ_m is first class if for all . Otherwise it is called second class. Second-class constraints χ_k indicate the presence of unphysical degrees of freedom, such that Dirac suggested to eliminate them by changing the Poisson bracket into the Dirac bracket , where C_ij is the matrix of the Poisson brackets of the second class constraints. Dirac's program was to quantize by means of the correspondence principle. Henneaux and Teitelboim (Reference Henneaux and Teitelboim1989) instead rewrite the second-class constraints as first-class constraints in order to circumvent an early choice of a specific representation. But in this case quantization most likely yields gauge anomalies that prevent us from obtaining the physical states by simply imposing the gauge condition . Already for the electromagnetic field, there are different gauge-fixing procedures that yield different classifications of constraints, which accordingly imply different routes of quantization.

Dirac's original formulation of the conjecture was careful and far less precise than mathematical conjectures typically are: “I think it may be that all the first-class secondary constraints should be included among the transformations which don't change the physical state, but I haven't been able to prove it. Also I haven't found any example for which there exist first-class constraints which do generate a change in the physical state” (Reference Dirac1964, 23–24). Today, one knows a couple of counterexamples, among them a free theory in which the phase factor ϕ is considered a dynamical variable. Mathematical physicists have also found conditions under which Dirac's conjecture holds, among them if the reduction algorithm does not mix first- and second-class constraints and if all constraints can be uniquely attributed to a generation (Henneaux and Teitelboim Reference Henneaux and Teitelboim1989). The philosophically important points are that the theory of constraints is quite complicated already in the classical domain and that new challenges arise in the transition to quantum theory.

Ward Struyve has recently provided a manifestly gauge-invariant formulation of the Higgs mechanism, thus accomplishing the objectivation demanded by Earman or—as Struyve puts it—avoiding “the breakdown of determinism” (Reference Struyve2011, 230). Starting from the usual Lagrangian for a local U(1) symmetry

and taking up a proposal by Higgs (Reference Higgs1966), Struyve introduces a transformation by , . The new fields ρ, are gauge invariant, and θ transforms like a pure gauge variable , where corresponds to the unitary gauge used in the standard textbook formulation. Although singled out as the gauge degree of freedom, θ is not pure gauge because the gauge group must preserve the boundary conditions of the physical fields ϕ, A _μ, such that its constant value at infinity θ _∞ is in effect a physical variable. In these new gauge-invariant quantities, one can pass to the Hamiltonian in the reduced phase space because breaking the residual gauge symmetry is trivial. “The disadvantage of this approach is that it employs a polar decomposition of the scalar field, which is only valid when the scalar field is nowhere zero. Therefore, we present a different way to pass to reduced phase space, which does not pose such a restriction” (Struyve Reference Struyve2011, 234). This strategy invented by Dirac (Reference Dirac1955), however, in some key aspects, resembles the standard textbook argument and involves a perturbative expansion in terms of small fields, one of which is identified as a scalar field. While at this point one can apply the standard methods of quantization, including their ambiguities, Struyve's manifestly gauge-invariant theory—if intended to satisfy the demand for an objective account in general—poses bigger quantization problems than merely to guarantee that . For there simply is no canonical quantization in polar coordinates , which would be required if one wants to keep both ρ and θ as dynamical variables and simply employ the correspondence principle.

Thus, it seems fair to say that Struyve has succeeded in implementing Earman's program in classical field theory, while its quantum part remains unresolved. Yet to my mind the quantum theory represented the principal motivation of Earman's program, not least because he considered the Higgs mechanism as a test case for AQFT itself.

In the remainder of the article, I will cite three results obtained in the setting of AQFT. They indicate that a definitive answer to the first tier might not be a simple yes/no alternative and provide a new look at Goldstone's theorem classifying the possible cases. They also provide some evidence that progress along the lines of Earman's program will most likely be slow and require its opportunistic employment.

3. The Goldstone Theorem in AQFT

In a paper largely neglected by philosophers, Buchholz et al. (Reference Buchholz, Doplicher, Longo and Roberts1992) have given a detailed analysis of Goldstone's theorem. While in the textbook formulation, constraints and gauge invariances arise when working on the Lagrangian, AQFT is from the very beginning based solely on local observables, which are by definition invariant under gauge transformations, such that Earman's objectivity criterion is fulfilled once the appropriate net of algebras and the corresponding vacuum state have been specified. Yet in return, advocates of AQFT have to prove that the structures necessary to describe gauge symmetries can at all be “recovered from the observables” (Buchholz et al. Reference Buchholz, Doplicher, Longo and Roberts1992, 49). Indeed, in any theory fulfilling the Haag-Kastler axioms there exists a canonical field algebra with normal commutation (or anticommutation) relations that can be associated to the algebra of observables . The gauge group G acts on and describes those gauge symmetries that are not spontaneously broken and determines the irreducible representations. Yet G is only part of larger gauge group Γ that involves spontaneously broken symmetries but leaves the localization properties of the fields invariant, which justifies its interpretation as a gauge group in the first place. Since there exist operators that transform nontrivially under Γ, the local algebras are not maximal and Haag duality fails; that is, , with , being the spacelike complement and the commutant. The ‘missing’ local operators, which transform nontrivially under Γ but stay in the vacuum sector, generate an intermediate algebra . From the standpoint of AQFT, whose ontology is based on the net of local algebras, one may even consider this relation as a more suitable expression of the phenomenon, or rather possibility, of SSB. For, inclusions of algebras are a standard method within the algebraic approach.

Buchholz et al., however, start from the standard definition, whose clarity had been praised by Earman. “The gauge symmetry is said to be spontaneously broken if there is no unitary operator implementing it in the (irreducible) vacuum representation of or, what amounts to the same thing, if the vacuum state is not stable under the action of γ” (Buchholz et al. Reference Buchholz, Doplicher, Longo and Roberts1992, 50). More formally, if is the vacuum state of and , , the generator of a one-parameter group of automorphisms of , then the symmetry is spontaneously broken if .

The main result of their paper is achieved by finding an a priori upper bound of this term and studying the behavior of as the double cone expands to , s denoting the number of spatial dimensions. It turns out that there are four possible cases:

1. 1. If the theory has a mass gap, SSB only happens if grows exponentially with R.

2. 2. If grows like R^k with , then SSB occurs only in theories without a mass gap.

3. 3. If grows like R^k with , SSB is only possible in the case of a particle of mass zero; that is, in this case Goldstone bosons necessarily exist.

4. 4. If grows like R^k with , then there is no SSB whatever the mass spectrum is.

Even though there has not been much progress in extending the analysis to the Higgs mechanism, this classification is quite in the spirit of Earman's second tier and shows the virtues of an axiomatic analysis in the critical mode. What one sees more specifically is that the locality properties of the gauge symmetries are decisive, which is perhaps not too surprising for a theory of local algebras.

4. Dirac States, QED, and a Simple Higgs Model

In a series of papers, Hendrik Grundling and C. A. Hurst (Reference Grundling and Hurst1985, Reference Grundling and Hurst1987, Reference Grundling and Hurst1988a, Reference Grundling and Hurst1988b, Reference Grundling and Hurst1988c) have integrated Dirac's quantum theory of constraints into AQFT and studied the example of quantum electrodynamics (QED). The constraints, in this setting, do not emerge in a transition from the classical Lagrangian to the classical Hamiltonian but are put in from the very beginning as state conditions. Since classical phase space functions correspond to quantum observables, the transition to the reduced phase space should correspond to the transition to a reduced algebra of physical observables . The constraint algebra is defined as the subalgebra of the field algebra , whose members are in the kernel of the Dirac states, that is, the states that fulfill the constraints. Executing this program proves no less tricky than in the classical case.

Assuming that all physical information is contained in and a distinguished set of states Σ, Grundling and Hurst express the k constraints as one-parameter groups and define the Dirac states as the family of states ω on that fulfill the constraints; that is, , which for vector states equals . If we redefine the generator as and denote the subalgebra spanned by them as , it follows that . If , it can be shown that Dirac states exist and that the theory automatically contains only first-class constraints. These have again an algebraic structure, so that one can ultimately define , which is the smallest C∗ subalgebra in and its commutant . The concept of the weakly vanishing Poisson bracket is realized as the weak commutant . On can then prove another inclusion of algebras . If the Dirac conjecture holds, that is, if the first-class constraints generate the gauge transformations, describes the gauge-invariant quantities and those that are weakly gauge invariant. If one finally defines the algebra of physical observables as , one finds that it may fail to be simple, such that not all representations are faithful (), and the search for a maximal subalgebra for which this is the case might not succeed in each case.

To my mind, the complexity of this picture matches the standard constraint algorithm. Since the algebraic theory for second-class constraints is pretty clumsy, a proof of the Dirac conjecture in AQFT seems not in sight either. What can be proven, however, is that no Dirac state is regular; that is, if the algebra is represented by Weyl operators they are not strongly continuous such that one cannot recover operators corresponding to the classical quantities by differentiating the Weyl operators. This shows that—while in the presence of constraints the algebraic structure remains intact—Dirac states are pretty special and that in the presence of constraints Bohr's correspondence principle cannot be satisfied in a straightforward manner. This is, to my mind, the most important insight of implementing the Dirac theory in the algebraic approach, not a failure of the algebraic formalism.

It is not an exclusive feature of the Higgs mechanism that constraints cannot be implemented as operator identities. This is already the case for the electromagnetic field in the Lorentz gauge, where the constraint can only be implemented on all physical states by using the indefinite Gupta-Bleuler metric, a procedure that after quantization yields ghost states in the larger space that have to be removed by hand. The advantage of the Lorentz gauge is that Goldstone bosons exist and that one can thus study the interaction of both steps of the Higgs mechanism in a gauge-invariant setting.Footnote ²

If one works within the framework of AQFT and admits nonregular states, the Gupta-Bleuler metric can be avoided and the constraints imposed on the vacuum state. Walter Thirring and Heide Narnhofer (Reference Thirring and Narnhofer1992) have shown this at the example of an Abelian Higgs model that is coupled to the electromagnetic field, which accordingly acquires a mass due to the interaction with the Higgs field and without breaking any symmetry. The model defines and fixes such that it is simpler than Struyve's, which keeps ρ as an independent variable (his θ corresponds to S). The transformation yields a new Lagrangian:

The Lorentz condition is kept as a dynamical variable that will later be gauged away trivially by putting it to 0. Also S is pure gauge, but in virtue of the classical Euler-Lagrange equations it fulfills , S will be the variable corresponding to the constraint. In terms of these variables, one can define the Weyl operator

where are smooth functions over and define a multiplication law

and a gauge-invariant nonregular state

where are the usual invariant Green's functions and is the positive frequency part of Δ, while denotes a combination of second-order derivatives of . All this is standard in dealing with the electromagnetic field and of no specific relevance to the Higgs model.Footnote ³ Due to the nonregularity of the state , the field S does not correspond to an operator.

5. Conclusion

Although the model of by Thirring and Narnhofer is quite simple, it shows that a Higgs field can be treated as a constraint and implemented into a gauge-invariant nonregular state. The price for fulfilling Earman's objectivity requirement is, however, that the gauge-variant constraints do not exist as operators but only as Weyl operators. This poses problems if one wants to relate the classical and the quantum theory simply by Bohr's correspondence principle. If one instead considers quantum theory as the more basic theory, as most advocates of AQFT do, the Higgs mechanism may correspond to a genuine quantum phenomenon or, at least, require a formulation that is not the quantization of a classical Higgs model. The algebraic approach brings not much relief for the Dirac conjecture and his quantization algorithm; rather, the appearance of nonregular states indicates that their quantum analogues are rather peculiar. All these observations, largely obtained within the framework of AQFT, do not require that AQFT in its present form ultimately succeeds in integrating the SM. Yet they underscore its critical potential.