# How does a gauge field lead to charge superselection?

How does the electromagnetic gauge field lead to charge superselection and why does this fail when the symmetry is broken as e.g. in superconductors?

## Answers and Replies

How does the electromagnetic gauge field lead to charge superselection and why does this fail when the symmetry is broken as e.g. in superconductors?
I can't answer your question in detail, but I have been reading much more of Strocchi's work; it was more interesting than I had thought at first, it is basically rigorous (except when he draws verbal conclusions, where he is less careful), though his assumptions are not always to my taste. But he has a valid overall message, gained mainly through the analysis of simpler, exactly solvable models in 2 dimensions.

Superselection happens whenever a field can have nontrivial behavior at infinity. The C^* algebra of bounded observables then has multiple physically relevant inequivalent representations, each one corresponding to a superselection sector, and indexed by a corresponding charge. In a gauge theory, these charges are associated with the charge operators defined by Noether's theorem. If the symmetry is broken, the charges also go away (as my limited understanding tells me).

The charge structure means that the Wightman view only works for the charge zero part of the algebra, consisting of all gauge invariant observables. These are represented as self-adjoint operators, so that one can form expectations values of products. On the other hand, gauge-dependent observables are needed to create charged states, but since these are in different superselection sectors, they correspond to inequivalent representations, which means that these must be implemented differently. This is done through nonregular representations of the Weyl algebra, which leads to nonseparable Hilbert spaces of the same kind Kibble used already to address the infrared problem of QED.

I found Section 4 of Strocchi's paper http://arxiv.org/pdf/hep-th/0401143 a quite readable summary.

===============================================

For the sake of new readers, I collect below a few pieces of related discussion from two older threads:

it is well known that Wightman axioms are very difficult to satisfy, and actually impossible in gauge field theories (http://arxiv.org/abs/hep-th/0401143).
You over-interpret the paper. There are 2-dimensional gauge theories (e.g., the Schwinger model) satisfying the Wightman axioms. In 4D, there is not a single theorem against the existence of interacting Wightman fields; it is just that we currently lack the mathematical tools to decide either way.

Nothing excludes gauge fields since there is no agreed-upon way how to formulate the requirement of gauge invariance in the Wightman setting. If one formulation can be proved to lead to nonexistence, it only rules out this formulation as good.
Quote from http://arxiv.org/abs/hep-th/0401143: [Broken]

"In conclusion, quite generally one can prove that in the quantization of gauge field theories the (correlation functions of the) charged fields cannot satisfy all the quantum mechanical constraints QM1, QM2 and the relativity constraint R1, R2, since locality and positivity are crucially in conflict. Therefore, the general framework discussed in Sect.3.3 has to be modified (modified Wightman axioms)"

Maybe the paper is wrong, but certainly I do not over-interpret it.
I'd have said that you over-interpret the evidence given in the paper.

If one reads section 4 and look at what precedes this statement on p.23, one finds that the author doesn't give a proof. Reference (28) lists two sources,both by the author himself (already not a good sign), and seems to contain the evidence. The Phys Rev paper starts off with ''... standard QFT can be formulated in terms of fields satisfying all the standard axioms (positivity included)'', hence shows that he doesn't work on the rigorous level - since none of the standard QFTs in 4D has been shown rigorously to satisfy these axioms.
I don't have access to the book, but don't expect a higher level of rigor there.

Since nobody understands the IR problem for nonabelian gauge theories, let alone is able to prove anything about them rigorously 9in a positive or negative direction), his arguments are nothing more than plausibility considerations. And his conclusions are not shared by many. (Vienna, where I live, is the host of the Erwin Schroedinger Institute for Mathematical Physics; so I am informed first hand....)

There is even a 1 Million Dollar price for showing that 4D Yang Mills theory (the simplest nonabelian gauge theory) exists in the Wightman sense!
don't care; there are not so many physicists able to understand Strocchi ...
Well, I don't understand everything but at least some things in his papers.

Strocchi echoes (in many of his papers) more or less your arguments about why under certain assumptions the Gauss law implies the absence of charged states, but puts them into the framework of axiomatic field theory (where the completely different notation and terminology makes things look very different). This results in theorem that precisely specify the assumptions that go into the conclusions.

I started to look into the evidence Strocchi referenced [in the article mentioned by bg032]. I am still reading, and hope to present my findings later in this thread.

His formal exposition is generally rigorous (if one ignores somewhat looser talk in the introductions), but I find fault with his informal conclusions, since they are based on interpreting assumptions (stated in his theorems) that are far from trivial and by no means only formal translations of properties necessary for the real thing.

In particular, at present I don't think his no-go theorems are relevant for theories (like QCD) expected to have a mass gap. The situation may be different (i.e., not of Wightman form) for theories like QED that have massless asymptotic fields, because then the asymptotic states carrying the scattering physics cannot be described in a Fock representation but need more general coherent representations of the CCR (and different such representations for asymptotic states of different velocity).

The article mentioned by bg032 is about the latter situation.

A superselection sector is essentially an orbit of the algebra of local observables on a representing Hilbert space. (This is usually expressed by saying that superpositions between different superselection sectors are forbidden. See, e.g., http://en.wikipedia.org/wiki/Superselection) Thus it characterizes the asymptotic structure of a theory, capturing in particular the boundary conditions at infinity that we had been discussing.[/QUOTE]
In case anybody is wondering Strocchi's theorem proves a very restricted statement. In QED we have the Electron fields and the Photon Fields. Strocchi shows that if you assume:
(a) $$\Psi$$ and $$A_{\mu}$$ are Wightman Fields.
(b) They obey Maxwell's equations
(c) They are covariant
(d) Gauge invariance holds.

Then physical charge vanishes.

However this isn't really a problem. What the theorem is "really" proving is that there are no gauge invariant, local, covariant Wightman fields. QED can be described in terms of Wightman fields which are not covariant or in terms of Wightman fields which describe the dynamics directly, the "physical fields", instead of the Lagrangian fields. The problem with these is that non-covariant fields lose manifest covariance. The physical fields would make the Hamiltonian look hideous and you can't see gauge-invariance (similar to describing QCD with proton fields, the Hamiltonian would be infinitely long and gauge invariance of quarks and gluons would be invisible). You could also work with non-local objects like "Wilson loops" whose algebra would satisfy the Haag-Kastler axioms, but this would be even more difficult.

This is a problem for perturbation theory where you would like a covariant, local field for doing calculations. So if you want to do that you need to drop some assumption. Commonly we drop (b) and obtain an enlarged Hilbert space of states on which Maxwell's equations do not hold. In some subspace of this space they hold, the physical Hilbert space. This subspace is then specified by the Gupta-Bluer condition. (In Yang-Mills theories the interaction makes the condition more subtle and you need to enlarge the Hilbert space even further to obtain a simple linear condition. The correct enlargement is to include fermions with incorrect statistics, which you will know as ghosts.)

So we perform calculations in this enlarged Hilbert space, where we are allowed use a local, covariant field and compute physical state -> physical state processes.

Of course one could just calculate in the physical Hilbert space, but standard perturbation theory would be impossible, but the Wightman axioms do hold for the physical fields.

If anybody is afraid of the lack of rigour here, since I assume QED exists*, just pretend I am talking about QED in 2,3 dimensions where it does exist. Or consider the electron field to be classical in 4D. Or perhaps take my remarks in the Yang-Mills case.

*Which some doubt due to triviality.
I wonder why the assumption (a) is reasonable: Since A(x) is an unobservable, gauge-dependent field, I don't see any reason to suppose that it must be a Wightman field.
Both the requirement of causal commutation rules for A(x) and the requirement of Lorentz invariance for A(x) seem to be not gauge covariant, hence can hold, if at all, only in special gauges. But both are part of the assumption that A(x) is a Wightman field.

Is there anything left from Strocchi's assertions in his many papers on the subject if one drops these two assumptions?
in a sense Strocchi's theorem isn't really that surprising. Even Strocchi himself in some of his books makes this point, also see the book by Steinmann "Perturbative QED and Axiomatic Field Theory".

Strocchi is mainly concerned with issues that arise in a rigorous study of gauge theories that don't occur in other field theories. For example the theorem above simply shows that $$A_{\mu}$$ isn't a Wightman field so a rigorous treatment will not be as straight forward. Theorems like the above are also used to show where certain objects from formal field theory orginate from in a rigorous approach. So Strocchi and others such as Nakanishi show that the Gupta-Bluer condition and ghosts arise from trying to work with a field as "Wightman-like" as we can manage.

I don't think Strocchi is really pointing anything out, more just showing where naïve assumptions from formal field theory go wrong and what is really going on behind the scenes.
I would like to learn more about the connection between gauge fields and charge superselection; a problem which has turned up in this thread, too. However I don't want to hijack it. Therefore I started a new one in the quantum theory forum:
How does a gauge field lead to charge superselection?
As yet I got no reply to my posting #111, I went on reading and think I found some explanations which are nicely in line with the current discussion.
@article{wightman1995superselection,
title={{Superselection rules; old and new}},
author={Wightman, AS},
journal={Il Nuovo Cimento B (1971-1996)},
volume={110},
number={5},
pages={751--769},
issn={0369-3554},
year={1995},
publisher={Springer}
}

and

@article{strocchi1974proof,
title={{Proof of the charge superselection rule in local relativistic quantum field theory}},
author={Strocchi, F. and Wightman, A.S.},
journal={Journal of Mathematical Physics},
volume={15},
pages={2198},
year={1974}
}

The first article by Wightman is an easy read also for the non-specialist in field theory (like me) while the second one is highly technical.
The basic argument ( as far as I understood it) is that a global gauge symmetry leads to the existence of a conserved charge. If the gauge symmetry is furthermore local, this does not lead to any new conserved quantity but the charge current vector can be written as $$j^\mu=\partial_\nu F^{\mu \nu}$$ (forgive me potential sign errors) which encompasses Gauss law for the charge density.
Now as we already discussed Gauss law allows to express the charge inside a volume to be expressed in terms of the electric field on the boundary. But the electric field on the boundary will commute with all operators localized inside the region. Hence the charge commutes with all local operators which and is thus a classical quantity. That is precisely the statement of supersymmetry. In formulas:

$$\int dV [\rho(x), A]=\int dV [\text{div} E(x), A]=\int dS\cdot [E, A]=0$$
where A is any (quasi-) local operator.

Now this argument is not precise as Gauss law does not hold as an operator equation. Hence in the second article Strocchi and Wightman use the Gupta Bleuler formalism.
The argument still assumes that the total charge can be represented as a unitary operator. This statement breaks down if the symmetry is broken.

After Goldstones theorem there was a lot of discussion how it can be avoided leading eventually to the Higgs mechanism. As far as I understand, the condition for Higgs mechanism to apply coincide with the presence of a superselection rule in the unbroken case.

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Maybe I should first concretize my initial question. I am not specially interested in electromagnetism, i.e. symmetry breaking involving massless particles, which seems to be the difficult part of the problem.
As far as I understand it right now, it has been one of the great successes of aqft to derive the gauge symmetry of the field algebra, and the statistics of the particles, from the "observable" superselection sector structure and the algebra of quasi-local observables.
One assumption in this analysis is Haag's duality, which states, as I understood it, that the local observables (or their representation) on regions which are space like form all the commuting operators which commute with the quasi-local operators in a double cone. (Please feel free to comment on this. I am not sure that I understand correctly a statement like A(O)'=A(O') ).
From causality one can only conclude that the operators in space like separated regions form a sub set of all commuting operators. In fact, in situations with a broken symmetry, there seem to exist other commuting operators. I would be glad to see some concrete example.
There seems to be an interesting discussion of this topic which, however, is not accessible to me:
J. Roberts, “Spontaneously broken gauge symmetries and superselection
rules” in Proceedings of the International School of Mathematical Physics,
Camerino, 1974, ed. by G. Galavotti,1976

As far as I understand it right now, it has been one of the great successes of aqft to derive the gauge symmetry of the field algebra, and the statistics of the particles, from the "observable" superselection sector structure and the algebra of quasi-local observables.
I understand the abstract C^*-theoretic QFT only on a superficial level. But I can explain something of what happens in more ordinary terms for a concrete example:

The C^* algebra contains only gauge invariant field operators (or rather their exponentials). This means that charged field operators are not represented.
For the case of QED in the limit e-->0, this means that the generating operators are smeared versions of F=dA and of products of an electron field and a positron field.
The task (and the achievement) is to reconstruct from that the Fermion fields. This is done by finding the outer automorphism group of the C^* algebra - it contains all inner automorphisms (conjugation with exponentials) as a normal subgroup; the factor group is the global gauge group, in this case the abelian group generated by the electron charge.
Shifting the Fock representation (which describes the charge 0 structure) by a representative in the gauge group, of charge Q gives an inequivalent representation, which describes the charge Q sector of the theory. Augmenting the algebra by the charges, one gets a representation in a bigger space, the direct sum of all these representations, which is in this case the free Fock space. This is a simplified picture since it ignores the part played by the e/m field. But I don't fully understand the details yet....
There seems to be an interesting discussion of this topic which, however, is not accessible to me:
J. Roberts, “Spontaneously broken gauge symmetries and superselection
rules” in Proceedings of the International School of Mathematical Physics,
Camerino, 1974, ed. by G. Galavotti,1976
I don't have access to this paper either, but here are some related ones:

H. Joos and E. Weimar
On the covariant description of spontaneously broken symmetry in general field theory
http://cdsweb.cern.ch/record/874059/files/CM-P00061555.pdf

F. Strocchi,
Spontaneous symmetry breaking in local gauge quantum field theory; the Higgs mechanism,
Communications in Mathematical Physics 56 (1977), 57-78
(freely accessible through project Euclid)

Thank you for the links. The first one seems to be quite interesting while the second seems to be more about specific problems with massless particles and indefinite fields.

What I like especially in the axiomatic approach is to start from observables only and, as a substitute for the gauge field, the concept of superselection sectors, which are observable in so far as no one has observed a superposition of different charge states. The gauge fields then appear basically as a means to eliminate phase factors in superpositions of states from different superselection sectors by averaging over group action, right?
In a free theory, I only have to consider the vacuum and a state with one charge unit, the rest follows from tensor products. That statistics drops out from this is also not unexpected as the irreducible representations of unitary groups can be labeled by Young diagrams.
What is more unclear to me is where relativity and causality come into play as
broken symmetry also appears in non-relativistic systems, even broken gauge symmetry as in the case of a superconductor. I suppose in non-relativistic problems the equivalent of causality is that one has to assume that local observables with different localization commute at equal time.

What I like especially in the axiomatic approach is to start from observables only and, as a substitute for the gauge field, the concept of superselection sectors, which are observable in so far as no one has observed a superposition of different charge states. The gauge fields then appear basically as a means to eliminate phase factors in superpositions of states from different superselection sectors by averaging over group action, right?
As I understand it, the gauge fields are unobservable field operators which are formally generators of 1-parameter groups in a bigger C^*-algebra canonically constructed from the observable C^* algebra, but not realizable in a representation of the direct sum of the superselection representations (which can be done formally but then leads to IR divergences). But they are needed to go from one sector to the others. Without the gauge fields, the superselection sectors would be completely disconnected.
In a free theory, I only have to consider the vacuum and a state with one charge unit, the rest follows from tensor products. That statistics drops out from this is also not unexpected as the irreducible representations of unitary groups can be labeled by Young diagrams.
Bose or Fermi statistics drops out only in dimension >=4. In dimension 3 one can have anyon statistics (since spin is not quantized), and in dimension 2 the concept of statistics is meaningless since because of the phenomenon of bosonization (boson current from fermion field) and fermionization (fermion soliton created in boson representations)
The latter is related to the superselection structure, but I don't understand yet how.
What is more unclear to me is where relativity and causality come into play as
broken symmetry also appears in non-relativistic systems, even broken gauge symmetry as in the case of a superconductor. I suppose in non-relativistic problems the equivalent of causality is that one has to assume that local observables with different localization commute at equal time.
Yes, but I think the nonrelativistic, Galilei invariant case is far less constraining.

Unfortunately, I don't really know much more about this in the context of your question.

I am still struggling to understand this topic. Several articles (see below) on symmetry breaking state that symmetry is broken if from Noether's theorem we get some conserved current $$j_\mu$$ but the charge operator $$\lim_{V \to \infty} \int dV j_0$$ does not exist (because the limit doesn't converge).
Wouldnt that mean that in any state with a non-zero homogeneous charge density j_0 the U(1) symmetry leading to that charge is broken?
However, I have never heard that in a homogeneous non-interacting Fermi gas U(1) symmetry is broken.

1. R. F. Streater, „Spontaneous breakdown of symmetry in axiomatic theory“, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 287, Nr. 1411 (1965): 510.
2. H. Reeh, „Symmetry Operations and Spontaneously Broken Symmetries in Relativistic Quantum Field Theories“, Fortschritte der Physik 16, Nr. 11 & 12 (1968): 687-706.

I am still struggling to understand this topic. Several articles (see below) on symmetry breaking state that symmetry is broken if from Noether's theorem we get some conserved current $$j_\mu$$ but the charge operator $$\lim_{V \to \infty} \int dV j_0$$ does not exist (because the limit doesn't converge).
Wouldnt that mean that in any state with a non-zero homogeneous charge density j_0 the U(1) symmetry leading to that charge is broken?
However, I have never heard that in a homogeneous non-interacting Fermi gas U(1) symmetry is broken.
In infinite dimensions (and hence in QFT) there is a difference between representations of a group and representations of their Lie algebra. The near 1-1 correspondence familiar from the finite-dimensional case is no longer given, because of topological problems.

The nonexistence of the charge operator therefore does not imply the nonexistence of the U(1) representation. It only means that in the relevant representation, the U(1) has no infinitesimal generator (which would be the charge).

For background of such nonregular representations in the simplest case where the group is the Weyl group, see
Acerbi, F. and Morchio, G. and Strocchi, F.,
Infrared singular fields and nonregular representations of canonical commutation relation algebras,
Journal of Mathematical Physics 34 (1993), 899

Thanks for the interesting article. However as far as I understand it, the effects described are more typical of low dimensional systems.
Regarding the connection between representations of U(1) and its generator Q, isn't the relation still given by Stones theorem?
So if Q does not exist, the operations from U(1) are not unitarily implementable.

Thanks for the interesting article. However as far as I understand it, the effects described are more typical of low dimensional systems.
No. these effects are universal, they are _always_ present in case of IR divergences. They can be studied best in cases where the models are explicitly solvable, which is in 2D. But if you google for other papers of Morchio and Strocchi (say), you'll find some that look at infrared aspects of QED amd QCD.

Regarding the connection between representations of U(1) and its generator Q, isn't the relation still given by Stones theorem?
So if Q does not exist, the operations from U(1) are not unitarily implementable.

Indeed. Stone's theorem only applies for regular representations of the CCR.

No. these effects are universal, they are _always_ present in case of IR divergences. They can be studied best in cases where the models are explicitly solvable, which is in 2D. But if you google for other papers of Morchio and Strocchi (say), you'll find some that look at infrared aspects of QED amd QCD.

Hm, but at the end of page 913 he shows that in the model he considers non-regular reps only occur in dimensions 1+1 and 2+1, but not in 3+1.
Can you translate the mathematical definition of a "nonregular" representation into human language?
The definition of the sigma(F,G) seems to be some smeared out version of Heisenbergs commutation relation, but what is the "nondegenerate quadratic Hilbert form" q(.) and what is the consequence of it being infinite?

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Hm, but at the end of page 913 he shows that in the model he considers non-regular reps only occur in dimensions 1+1 and 2+1, but not in 3+1.
This statement applies for the particular Stueckelberg--Kibble-model only, not in general.
The example with the Zak transform is non-regular in arbitrary dimensions.
Can you translate the mathematical definition of a "nonregular" representation into human language?
Actually this paper is the very best paper in the literature in terms of explaining things in the context of intelligible examples; so if you don't understand it yet you'd really spend the time pondering the examples in detail. The examples tell what the construction ''means'' -- beyond that very little can be said in nontechnical terms.

The nonregular representation encodes a direct sum of lots of inequivalent irreducible representations, forming a nonseparable Hilbert space. This is the same Hilbert space also considered by Kibble and Kulish/Faddeev in the context of resolving the IR problem of QED (where everything is much messier).
The definition of the sigma(F,G) seems to be some smeared out version of Heisenbergs commutation relation, but what is the "nondegenerate quadratic Hilbert form" q(.) and what is the consequence of it being infinite?
q is infinite outside some subspace, so this is just saying that coherent states in the representation are orthogonal when their difference doesn't have finite norm - i.e., if the transformation mapping one to the other is not unitarily implementable.

You can work out easily the meaning of q and sigma by looking at the case of n independent oscillators and working out the form of the Weyl relations. Of course, this gives a regular representation, but it tells the ''standard'' form of sigma and q.

The main novelty of their paper is that by allowing q to be defined only on a subspace one gets lots of additional nonregular representations, and as his long list of example shows, these encode important problems. Each one shows a different facet of the meaning of the concept.

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From what I understand that all looks suspiciously like kind of the Wagner Mermin theorem from solid state physics.

From what I understand that all looks suspiciously like kind of the Wagner Mermin theorem from solid state physics.

Of course the results in Mermin and Wagner, Phys. Rev. Lett. 17, 1133 (1966), are IR phenomena, and hence are related. But their arguments apply only to two dimensions.

Many of the examples in Acerbi et al are 2-dimensional, since these are more tractable. But the ideas and the necessity and usefulness of nonregular representations are general, and are present whenever there are unitarily inequivalent representations related by a group of automorphisms of the observable algebra.

Dear Arnold,
I decided to understand the DHR analysis in somewhat more detail and read the chapter IV in R. Haag, Local quantum physics. I have some problem understanding the cross product of intertwiners (IV.2.15).
He defines charge operators $$\rho (A)=V\pi(A)V^{-1}$$ where A is an arbitrary observable, and V is a unitary operator, pi designs a representation of the algebra in a given Hilbert space.
Obviously, charge operators can be composed. He now introduces intertwiners R as $$(\rho'A)R=R(\rho A)$$ and designs them as $$\mathbf{R}=(\rho',R,\rho)$$.
Now he introduces a cross product of intertwiners $$\mathbf{R}_2 x\mathbf{R}_1=(\rho'_2\rho'_1, R_2\rho_2R_1,\rho_2\rho_1)$$, saying that it can easily be checked. I don't see how!

I decided to understand the DHR analysis in somewhat more detail
It would be nice if understanding were an act of decision. We can decide to study something, but understanding is a gift - though it often comes as the result of careful study.
and read the chapter IV in R. Haag, Local quantum physics. I have some problem understanding the cross product of intertwiners (IV.2.15).
He defines charge operators $$\rho (A)=V\pi(A)V^{-1}$$ where A is an arbitrary observable, and V is a unitary operator, pi designs a representation of the algebra in a given Hilbert space.
Obviously, charge operators can be composed. He now introduces intertwiners R as $$(\rho'A)R=R(\rho A)$$ and designs them as $$\mathbf{R}=(\rho',R,\rho)$$.
Now he introduces a cross product of intertwiners $$\mathbf{R}_2 x\mathbf{R}_1=(\rho'_2\rho'_1, R_2\rho_2R_1,\rho_2\rho_1)$$, saying that it can easily be checked. I don't see how!

I don't understand the DHR analysis myself; the C^* algebra language of the Haag school is too abstract for me, and I can interpret it only very superficially.

Also, I don't have Haag's book, and from what you write I can't guess what is going on. Maybe adding the definition of the composition of two rho's and two rhodash's would help.

In most cases when I consider posting a question in this forum, I find the answer while formulating the problem, as it forces me to formulate the question so that someone else can understand. This time it happened when trying to formulate an answer to your post.
The answer lies in Haag making a difference between the product of algebraic elements (e.g. A, R) and products of morphisms (rho) and how products of these quantities are to be interpreted (when using brackets etc ). I didn't quite understand this difference until now.
So
$$R_2 (\rho_2 R_1) \rho_2 \rho_1 A=R_2 V_2 R_1 V^{-1}_2V_1 A V^{-1}_1 V^{-1}_2$$. On the right hand side, only algebraic products appear and the calculation becomes trivial.
Nevertheless thank you very much.

In most cases when I consider posting a question in this forum, I find the answer while formulating the problem, as it forces me to formulate the question so that someone else can understand. This time it happened when trying to formulate an answer to your post.
The answer lies in Haag making a difference between the product of algebraic elements (e.g. A, R) and products of morphisms (rho) and how products of these quantities are to be interpreted (when using brackets etc ). I didn't quite understand this difference until now.
So
$$R_2 (\rho_2 R_1) \rho_2 \rho_1 A=R_2 V_2 R_1 V^{-1}_2V_1 A V^{-1}_1 V^{-1}_2$$. On the right hand side, only algebraic products appear and the calculation becomes trivial.
Nevertheless thank you very much.

Yes. I had noticed that the product must mean different things for the different items, hence asked for its definition.

johnhunt3055
I have been looking for the exam paper of this lead and gauge problem, last time I cleared It but some things which are out of my course. I know this is lead training discussion but I really need to find out the question which can lead me to clear the notes on lead-gauge problem held by pius.