Elementary Particles: What are they? Substances, Elements and Primary Matter

Abstract

The extremely successful Standard Model of Particle Physics allows one to define the so-called Elementary Particles. From another point of view, how can we think of them? What kind of a status can be attributed to Elementary Particles and their associated quantised fields? Beyond the unprecedented efficiency and reach of quantum field theories, the current paper attempts at understanding the nature of what these theories describe, the enigmatic reality of the quantum world.

Appendix

In QFTs a non trivial case is that of condensates. More specifically, are condensates intrinsically related to vacuum transition amplitudes, $\langle 0_+|0_-\rangle$, such as depicted in Fig. 1 in the case of QED, or can they be obtained by relying on calculations of S-matrix elements like the Casimir effects?

Condensates being recognised physical realities in the QFT standard model context Jaffe (2005), if they can exclusively be calculated out of the vacuum energy density related to $\langle 0_+|0_-\rangle$, then this vacuum energy density could be regared as endowed with some physical reality also. This is the question at stake.

A famous and most important example is that of the dynamical chiral symmetry breaking mechanism in QCD or QED. A measure of this chiral symmetry breaking phenomenon is provided by the Lorentz scalar and gauge invariant fermionic condensate $\langle \bar{ \Psi }\Psi (x)\rangle$, and is therefore recognized as an order parameter of this symmetry. To simplify somewhat while preserving the point, in an abelian theory like QED this order parameter can be obtained by calculating the condensate according to the relation,

$$\begin{aligned} \langle {\bar{\Psi }}\Psi (x)\rangle =- \frac{1}{V} \frac{\partial }{\partial m}\,\ln \,\langle 0_+|0_-\rangle \end{aligned}$$

(18)

where m stands for a relevant fermionic mass and V for the overall spacetime volume. In this way, explicit reference is made to the series of vacuum diagrams of Fig. 1, related to the would-be vacuum energy density. In Ref. Fried and Grandou (1985) for instance, this equation was used successfully to evaluate the fermionic condensate in the massive Schwinger model (i.e., massive QED at two spacetime dimensions, non-integrable contrary to the massless version). But using (18) is in no way mandatory, as the same result could be achieved relying on a 4-point Green’s function calculation Guerin and Fried (1986), parts of which related to some scattering processes among particles.

More recently the same procedure has been successfully extended to QCD Grandou and Tsang (2019) avoiding the creation of a link of necessity between the condensate and a would-be non-zero vacuum energy density, as was questioned in Jaffe (2005). As well known connected Green’s function don’t even ‘see’ the pure phase factor $\langle 0_+|0_-\rangle$.

After all, that particles involved in selected configurations of scattering processes may ‘feel’ the presence of condensates at the energy scales where they come about seems quite plausible.

Discussing this point further falls beyond the scope of the present paper but it is worth signalling also that in the light front formulation of QCD, dynamical chiral symmetry breaking is not a property of the hadron-less vacuum state of QCD Casher and Susskind (1974). The questions in the conclusion of Jaffe (2005) could very well find a clue in that, as suspected by P. Carruthers .. it is a mistake to identify the ground state with the vacuum Carruthers (1997). The condensate calculations of Grandou and Tsang (2019) and Grandou and Hofmann (2015) can be seen as an illustration of this wisdom.