Editing Haag's theorem (section)

== Quantum field theorists’ conflicting reactions ==
While some physicists and philosophers of physics have repeatedly emphasized how seriously Haag's theorem undermines the foundations of [[quantum field theory|QFT]], the majority of practicing quantum field theorists simply dismiss the issue. Most quantum field theory texts geared to practical appreciation of the [[Standard Model]] of [[elementary particle]] interactions do not even mention it, implicitly assuming that some rigorous set of definitions and procedures may be found to firm up the powerful and well-confirmed heuristic results they report on.

For example, asymptotic structure (cf. [[Jet (particle physics)|QCD jets]]) is a specific calculation in strong agreement with experiment, but nevertheless should fail by dint of Haag's theorem. The general feeling is that this is not some calculation that was merely stumbled upon, but rather that it embodies a physical truth. The practical calculations and tools are motivated and justified by an appeal to a grand mathematical formalism called [[quantum field theory|QFT]]. Haag's theorem suggests that the formalism is not well-founded, yet the practical calculations are sufficiently distant from the abstract formalism that any weaknesses there do not affect (or invalidate) practical results.

As was pointed out by Teller (1997):<ref name=Teller-1997 /><blockquote>Everyone must agree that as a piece of mathematics Haag’s theorem is a valid result that at least appears to call into question the mathematical foundation of interacting quantum field theory, and agree that at the same time the theory has proved astonishingly successful in application to experimental results.<ref name=Teller-1997>{{cite book |last=Teller |first=Paul |year=1997 |title=An Interpretive Introduction to Quantum Field Theory |page=115 |publisher=Princeton University Press}}</ref></blockquote> Tracy Lupher (2005)<ref name=Lupher-2005 /> suggested that the wide range of conflicting reactions to Haag's theorem may partly be caused by the fact that the same exists in different formulations, which in turn were proved within different formulations of [[quantum field theory|QFT]] such as [[Wightman axioms|Wightman's axiomatic approach]] or the [[LSZ reduction formula|LSZ formula]].<ref name=Lupher-2005 /> According to Lupher,<blockquote>The few who mention it tend to regard it as something important that someone (else) should investigate thoroughly.<ref name=Lupher-2005>{{cite journal |last=Lupher |first=Tracy |year=2005 |title=Who proved Haag's theorem? |journal=International Journal of Theoretical Physics |volume=44 |issue=11 |pages=1993–2003 |doi=10.1007/s10773-005-8977-z |bibcode=2005IJTP...44.1995L|s2cid=120271840 }}</ref></blockquote>

[[Lawrence Sklar]] (2000)<ref name=Sklar-2000 /> further pointed out:<blockquote>There may be a presence within a theory of conceptual problems that appear to be the result of mathematical artifacts. These seem to the theoretician to be not fundamental problems rooted in some deep physical mistake in the theory, but, rather, the consequence of some misfortune in the way in which the theory has been expressed. Haag’s theorem is, perhaps, a difficulty of this kind.<ref name=Sklar-2000>{{cite book |last=Sklar |first=Lawrence |author-link=Lawrence Sklar |year=2000 |title=Theory and Truth: Philosophical critique within foundational science |publisher=Oxford University Press}}</ref></blockquote>

David Wallace (2011)<ref name=Wallace-2011 /> has compared the merits of conventional QFT with those of [[Local quantum field theory|algebraic quantum field theory (AQFT)]] and observed that<blockquote>... [[Local quantum field theory|algebraic quantum field theory]] has unitarily inequivalent representations even on spatially finite regions, but this lack of [[Self-adjoint operator|unitary equivalence]] only manifests itself with respect to expectation values on arbitrary small [[spacetime]] regions, and these are exactly those expectation values which don’t convey real information about the world.<ref name=Wallace-2011>{{cite journal |last=Wallace |first=David |year=2011 |title=Taking particle physics seriously: A critique of the algebraic approach to quantum field theory |journal=Studies in History and Philosophy of Science |series=Part&nbsp;B: Studies in History and Philosophy of Modern Physics |volume=42 |issue=2 |pages=116–125|doi=10.1016/j.shpsb.2010.12.001 |bibcode=2011SHPMP..42..116W |citeseerx=10.1.1.463.1836 }}</ref></blockquote> He justifies the latter claim with the insights gained from modern [[renormalization group]] theory, namely the fact that<blockquote>... we can absorb all our ignorance of how the cutoff [i.e., the short-range cutoff required to carry out the renormalization procedure] is implemented, into the values of finitely many coefficients which can be measured empirically.<ref name=Wallace-2011 /></blockquote>

Concerning the consequences of Haag's theorem, Wallace's observation<ref name=Wallace-2011 /> implies that since QFT does not attempt to predict fundamental parameters, such as particle masses or coupling constants, potentially harmful effects arising from [[Self-adjoint operator|unitarily non-equivalent]] representations remain absorbed inside the empirical values that stem from measurements of these parameters (at a given [[length scale]]) and that are readily imported into QFT. Thus they remain invisible to quantum field theorists, in practice.