Weyl's search for a difference between physical' and mathematical' automorphisms (original) (raw)
Related papers
A Perdurable Defence to Weyl’s Unified Theory
Journal of Modern Physics, 2014
Einstein dealt a lethal blow to Weyl's unified theory by arguing that Weyl's theory was at the very best-beautiful, and at the very least, un-physical, because its concept of variation of the length of a vector from one point of space to the other meant that certain absolute quantities, such as the "fixed" spacing of atomic spectral lines and the Compton wavelength of an Electron for example, would change arbitrarily as they would have to depend on their prehistories. This venomous criticism of Einstein to Weyl's theory remains much alive today as it was on the first day Einstein pronounced it. We demonstrate herein that one can overcome Einstein's criticism by recasting Weyl's theory into a new Weyl-kind of theory were the length of vectors are preserved as is the case in Riemann geometry. In this New Weyl Theory, the Weyl gauge transformation of the Riemann metric g µν and the Maxwellian electromagnetic field A µ are preserved.
Klein-Weyl's Program and the Ontology of Gauge and Quantum Systems
We distinguish two orientations in Weyl's analysis of the fundamental role played by the notion of symmetry in physics, namely an orientation inspired by Klein's Erlangen program and a phenomenological-transcendental orientation. By privileging the former to the detriment of the latter, we sketch a group(oid)-theoretical program—that we call the Klein-Weyl program—for the interpretation of both gauge theories and quantum mechanics in a single conceptual framework. This program is based on Weyl's notion of a "structure-endowed entity "equipped with a "group of automorphisms". First, we analyze what Weyl calls the "problem of relativity" in the frameworks provided by special relativity, general relativity, and Yang-Mills theories. We argue that both general relativity and Yang-Mills theories can be understood in terms of a localization of Klein's Erlangen program: while the latter describes the group-theoretical automorphisms of a single structure (such as homogenous geometries), local gauge symmetries and the corresponding gauge fields (Ehresmann connections) can be naturally understood in terms of the groupoid-theoretical isomorphisms in a family of identical structures. Second, we argue that quantum mechanics can be understood in terms of a linearization of Klein's Erlangen program. This stance leads us to an interpretation of the fact that quantum numbers are "indices characterizing representations of groups" (Weyl-1931, p.xxi) in terms of a correspondence between the ontological categories of identity and determinateness.
Reichenbach, Weyl, Philosophy and Gauge
Fundamental Theories of Physics, 2020
Hermann Weyl connected his epoch-making work on general relativity and gauge theory to his Husserlian views about the phenomenological essence of space and time. This philosophical stance of Weyl's has received considerable attention in recent years and has been favorably compared and contrasted with the "logicalempiricist" approach of Reichenbach, Weyl's contemporary who wrote extensively about relativity and the philosophy of space and time. We will argue, however, that Weyl's use of phenomenology should be seen as a case of personal heuristics rather than as a systematic and viable philosophy of physics. We will explain and defend Reichenbach's sophisticated empiricism, which in our opinion has often been misunderstood, and argue that it is better suited as a general philosophical framework for the natural sciences than Weyl's phenomenology. In the second half of the nineteenth and the beginning of the twentieth century Kant's original version of this "synthetic a priori" doctrine came under increasing
On the Failure of Weyl's 1918 Theory
2014
In 1918 the German mathematician Hermann Weyl developed a non-Riemannian geometry in which electromagnetism appeared to emerge naturally as a consequence of the non-invariance of vector magnitude. Although an initial admirer of the theory, Einstein declared the theory unphysical on the basis of the non-invariance of the line element ds, which is arbitrarily rescaled from point to point in the geometry. We examine the Weyl theory and trace its failure to its inability to accommodate certain vectors that are inherently scale invariant. A revision of the theory is suggested that appears to refute Einstein's objection.
Fundamentality, Effectiveness and Objectivity of Gauge Symmetries
International Studies in the Philosophy of Science, 2016
Much recent philosophy of physics has investigated the process of symmetry breaking. Here, I critically assess the alleged symmetry restoration at the fundamental scale. I draw attention to the contingency that gauge symmetries exhibit, i.e. the fact that they have been chosen from among a count-ably infinite space of possibilities. I appeal to this feature of group theory to argue that any metaphysical account of fundamental laws that expects symmetry restoration up to the fundamental level is not fully satisfactory. This is a symmetry argument in line with Curie's 1 st principle. Further, I argue that this same feature of group theory helps to explain the " unreasonable " effectiveness of (this subfield of) mathematics in (this subfield of) physics, and that it reduces the philosophical significance that has been attributed to the objectivity of gauge symmetries.
From Hermann Weyl to Yang and Mills to Quantum Chromodynamics
Nuclear Physics, 2005
This is a personal view of 1 the developments from the invention of the concept of gauge invariance to our present understanding that it provides the fundamental principle for the construction of theories of forces between the basic blocs of matter. This journey was full of twists and turns and marked by fascinating moments. It is these aspects of the development of gauge theories that I will concentrate on. Although Yang-Mills theories provide the basic framework for both strong and electroweak interactions, my contribution concerns almost exclusively the former only. 1 There are many excellent articles discussing various aspects of the development of Yang-Mills theories [1-3]. The contribution of Weyl toward the concept of gauge invariance is discussed in [4].
Same Diff? Part II: A compendium of similarities between gauge transformations and diffeomorphisms
2021
How should we understand gauge-(in)variant quantities and physical possibility? Does the redundancy present in gauge theory pose different interpretational issues than those present in general relativity? Here, I will assess new and old contrasts between general relativity and Yang-Mills theory, in particular, in relation to their symmetries. I will focus these comparisons on four topics: (i) non-locality, (ii) conserved charges, (iii) Aharonov-Bohm effect, and (iv) the choice of representational conventions of the field configuration. In a companion paper, I propose a new contrast and defend sophistication for both theories.
The British Journal for the Philosophy of Science, 2010
This article investigates an intertwined systematic and historical view on theories of matter. It follows an approach brought forward by Hermann Weyl around 1925, applies it to recent theories of matter in physics (including geometrodynamics and quantum gravity), and embeds it into a more general philosophical framework. First, I shall discuss the physical and philosophical problems of a unified field theory on the basis of Weyl's own abandonment of his 1918 'pure field theory' in favour of an 'agent theory' of matter. The difference between agent and field theories of matter is then used to establish a sort of dialectic meta-view. With reference to Weyl this view can be understood as being a particular Fichtean transcendental idealist approach which attempts to combine the strengths of the Husserlian phenomenology and Cassirer's neo-Kantianism.
2021
The following questions are germane to our understanding of gauge-(in)variant quantities and physical possibility: in which ways are gauge transformations and spacetime diffeomorphisms similar, and in which are they different? To what extent are we justified in endorsing different attitudes—sophistication, quidditism/haecceitism, or full elimination—towards each? In a companion paper, I assess new and old contrasts between the two types of symmetries. In this one, I propose a new contrast: whether the symmetry changes pointwise the dynamical properties of a given field. This contrast distinguishes states that are related by a gauge-symmetry from states related by generic spacetime diffeomorphisms, as being ‘pointwise dynamically indiscernible’. Only the rigid isometries of homogeneous spacetimes fall in the same category, but they are neither local nor modally robust, in the way that gauge transformations are. In spite of this difference, I argue that for both gauge transformations ...
Synopsis and Discussion: Philosophy of Gauge Theory
Richard Healey's talk was divided in two parts. In the first part he argued that we are not justified in believing that localized gauge potential properties are there, but we are in believing that holonomy properties are. In the second part, he conceded that the holonomy interpretation offers an incomplete local and causal account, but he maintained that the onus is on QM.