Hardy's approach to quantum gravity and QM interpretation

[kith]

Work in quantum foundations is partly considered important because of the hope that the way we think about QM may point to a road to quantum gravity. Lucien Hardy, who is well-known in quantum foundations for his reformulation of QM in terms of five "reasonable" axioms, is one of the people who try to make this really tangible.

A couple of weeks ago, he put a new preprint with the title The Construction Interpretation: Conceptual Roads to Quantum Gravity on the arXiv.

This sounds ambitious, and it is. The main theme of the paper is to learn the right lessons from the conceptual development of GR which combined Newtonian gravity with special relativistic field theory and apply them in order to discover QG.

He notes that GR as the solution to the problems with Newtonian gravity (action at a distance) looked nothing like the efforts of Newton and his successors to solve these problems. And he argues that even if they did manage to solve the problem from the inside, they would have only gotten Newton-Cartan theory which isn't of much physical interest. So instead of trying to solve the problem of QG from within the paradigm of either QM or GR, he argues for a more radical starting point. (String theory, for example, operates under the quantum paradigm and attacks the problem by making gravity quantum (please correct me if I'm wrong))

For people who are interested in the interpretation of QM, Hardy has a rather sobering message:

If Quantum Gravity requires a radical departure from existing theories, the most interpretations of Quantum Theory can aspire to is to be the correct limiting version of Quantum Gravity just as the Newton-Cartan formalism is the correct limit of General Relativity. [...] Few physicists wish to dedicate themselves to merely providing amusement to future historians. The point of The Construction Interpretation is to take the noble instincts that lead us to attempt to understand Quantum Theory in conceptual terms and re-purpose them to the problem of constructing a theory of Quantum Gravity.

His suggestion is to chop the process of the discovery of GR into 7+3 distinct conceptual steps and perform analogous steps in order to find QG. His steps involve a quite instrumentalist and unusual way of looking at things which I can't digest easily. But I have only superficial knowledge of QFT and GR, so I can't follow the details anyway.

Thoughts? It looks like a very rough sketch to me. But I can neither judge how sensible this research program is, nor how far it already got. In QM, he does cite quite a bit of research of which I wasn't aware.

 

[MathematicalPhysicist]

I thought to myself that a new theory is in order which is different from both QFT and GR.

But I don't know how to even start building such a theory.
I might give a look at his paper.

 

[atyy]

There is already one good suggestion for quantum gravity - AdS/CFT. In informal remarks, it has been said (not sure who, Arkani-Hamed, Polchinski maybe) that this is because quantum mechanics needs an observer, and AdS provide a space for the observer on the boundary. So perhaps it might be a good idea to see if we can generalize the notion of observer in QM (eg. Brukner, Hardy), or if we can get rid of the observer (measurement problem).

I'm not sure if Witten's remarks about observables in dS space are related: https://arxiv.org/abs/hep-th/0106109.

There is also interesting work in AdS/CFT about how quantum mechanics arises in the bulk: https://arxiv.org/abs/1801.08101.

Personally, I'd like to see someone try to develop Bohmian AdS/CFT.

 

[Demystifier]

Personally, I'd like to see someone try to develop Bohmian AdS/CFT.

As far as I know, the only person in the world who might be crazy enough* to try to do something like that is - me.
But the problem is that I don't really buy AdS/CFT in the usual form, for the reasons I explained in https://lanl.arxiv.org/abs/1507.00591 . In short, I buy that boundary can be reproduced from the bulk, but I don't buy the converse, that the bulk can be reproduced from the boundary.

*By "crazy enough", I mean in the sense of Bohr's famous "Your theory is crazy but not crazy enough".

 

[MathematicalPhysicist]

As far as I know, the only person in the world who might be crazy enough* to try to do something like that is - me.
But the problem is that I don't really buy AdS/CFT in the usual form, for the reasons I explained in https://lanl.arxiv.org/abs/1507.00591 . In short, I buy that boundary can be reproduced from the bulk, but I don't buy the converse, that the bulk can be reproduced from the boundary.

*By "crazy enough", I mean in the sense of Bohr's famous "Your theory is crazy but not crazy enough".

What did Bohr mean?

literally or metaphorically?

 

[Demystifier]

What did Bohr mean?
literally or metaphorically?

Metaphorically, of course.

 

[MathematicalPhysicist]

Metaphorically, of course.

I don't know, QFT is quite a crazy theory...
literally.

 

[Demystifier]

I don't know, QFT is quite a crazy theory...
literally.

What do you find the craziest part of QFT?

 

[MathematicalPhysicist]

What do you find the craziest part of QFT?

The whole edifice, loads of calculations and nothing is truly rigorously mathematically justified.

At least with GR I can tell that it's really all about Riemannian Geometry.
But in QFT I find it hard to comprehend all the implications, and in Peskin and Schroeder it's hard to find how every equation is entailed from the one before.

If they had written a book that portrays every implication it would be something like 1000-2000 pages of calculations, too much to not find some error in it.

So to me, Bohr was dead serious about craziness.
And Peskin and Schroeder is only an introduction!

 

[atyy]

As far as I know, the only person in the world who might be crazy enough* to try to do something like that is - me.
But the problem is that I don't really buy AdS/CFT in the usual form, for the reasons I explained in https://lanl.arxiv.org/abs/1507.00591 . In short, I buy that boundary can be reproduced from the bulk, but I don't buy the converse, that the bulk can be reproduced from the boundary.

*By "crazy enough", I mean in the sense of Bohr's famous "Your theory is crazy but not crazy enough".

I need to read your explanation, but before that, I think there are many who would allow that it may not be a true duality, and that the boundary may serve as the definition of the bulk, but not the other way round.

 

[Demystifier]

I think there are many who would allow that it may not be a true duality, and that the boundary may serve as the definition of the bulk, but not the other way round.

Do you have some references?

 

[atyy]

Do you have some references?

These are all not exactly it, but maybe:

https://arxiv.org/abs/1007.4001
As a consequence of the last property, we consider such QFTs to be definitions of models of quantum gravity, with fixed asymptotic background. The idea that AdS/CFT defines a duality between two independently defined theories, is probably without merit.

https://arxiv.org/abs/1501.00007
While the AdS/CFT duality as such has not been rigorously ‘proved’ (partly because we do not yet have a complete independent definition of the quantum gravity side of the correspondence), ...

https://arxiv.org/abs/1609.00026
These realize the holographic principle directly: the quantum gravitational theories are defined as ordinary non-gravitational quantum theories (typically quantum field theories) on a fixed lower-dimensional spacetime

 

[Fra]

As far as I know, the only person in the world who might be crazy enough* to try to do something like that is - me.

You solipsist hidden variables are IMO an interesting research direction from someone in the bohmian field. I can relate to this idea, which is that "problem" of the hidden variables in ordinary bohmian mechanis is solves simply because they are not inferrable from other perspectives; but while the actual hidden structure EXPLAINS the interactions, but they remain probabilistic. So i do not think one will recover determinism, but the idea itself is interesting and it will unify both the world of uncertainy and realism, except the realism is observer dependent. This may sound like toying with words but i can envision how it actually makes sense.

I would enjoy reading any follow up papers you may have in this direction.

/Fredrik

 

[mitchell porter]

Personally, I'd like to see someone try to develop Bohmian AdS/CFT.

The immediate problem here is the need for gauge-fixing. I don't know any way to preserve gauge symmetry or general covariance in Bohmian mechanics. You have to just fix a gauge, as in Shtanov's Bohmian gravity. It's not very satisfying, though it would still be of interest to see it done for AdS/CFT. Perhaps one could start with a study of Shtanov supergravity in AdS.

 

[atyy]

The immediate problem here is the need for gauge-fixing. I don't know any way to preserve gauge symmetry or general covariance in Bohmian mechanics. You have to just fix a gauge, as in Shtanov's Bohmian gravity. It's not very satisfying, though it would still be of interest to see it done for AdS/CFT. Perhaps one could start with a study of Shtanov supergravity in AdS.

I was thinking Hamiltonian lattice gauge theory for the CFT side. I'm not sure what the status of lattice supersymmetry is.

 

[martinbn]

In short, I buy that boundary can be reproduced from the bulk, but I don't buy the converse, that the bulk can be reproduced from the boundary.

I may be confused by the language, because I don't even know what the statements of AdS/CFT are, but often the solutions to the equations in the whole region are determined by the values on the boundary. So, at least superficially it seems ok as a proposal (whatever the conjecture might be).

 

[Fra]

I suspect my ideas are even further from the original ambitions of Hidden variables, but i defend and enjoy the principal idea (although the principal idea may be implemented in various programs)

The immediate problem here is the need for gauge-fixing. I don't know any way to preserve gauge symmetry or general covariance in Bohmian mechanics. You have to just fix a gauge,

As I see it, one way of solving this is to accept that the need for gauge-fixing, and choosing an observer, is more fundamental and prior to desire to preserve gauge invariance and observer equivalence.

...the deterministic point-particle trajectories are associated only with the essential degrees of freedom of the observer, and not with the observed objects. In contrast with Bohmian HV's, nonlocality in solipsistic HV's can be substantially reduced down to microscopic distances inside the observer...

One possible solution then is to view gauge symmetry and observer consistency as emergent and NOT fundamental in the sense that all the different "hidden variables" in all the various obserers in the physical systems, with time, due to their interactions evolve to be in TUNE. Its the initial inconsistencies that constitute the evolutionary "forces" to deform the internal structure of observers. But the actual mechanism can be evolutionary self-organisation, so there are not non-local interactions as the inteactions are in the history.

But to implement this, requires a model for the "microstructure" of the observers, to see how they "negotiate" and interact. And probably requires this to be combined with something more radical. Closes thing that comes to mind from main programs is strings and the 6D manifolds. And maybe the selection of the manifolds in some landscape can be identified with the negotiation process. Then the string itself would be considered as an elementary observer. But the same idea could be tried with other microstructures as well.

/Fredrik

 

[Demystifier]

I may be confused by the language, because I don't even know what the statements of AdS/CFT are, but often the solutions to the equations in the whole region are determined by the values on the boundary. So, at least superficially it seems ok as a proposal (whatever the conjecture might be).

As the simplest example, consider the equation
$$\frac{d^2f(x)}{dx^2}+k^2f(x)=0$$
for $x\in [0,2\pi]$ and $k$ a real number. Let the boundary condition be $f(0)=f(2\pi)=0$. How that determines $f(x)$ for all $x$?

 

[martinbn]

Is that supposed to convince me that boundary conditions never determine the solution?

 

[Demystifier]

Is that supposed to convince me that boundary conditions never determine the solution?

No, it's supposed to convince people who know something about AdS/CFT that the statement that bulk is encoded in the boundary is a very nontrivial statement.

 

[kith]

There is already one good suggestion for quantum gravity - AdS/CFT.

Do you know how this fits into the framework with the 7+3 steps of Hardy's paper?

AdS/CFT is part of string theory and string theory is a quantum theory. So it operates mostly from within one of the two old paradigms which makes me suspect that it doesn't fit well into Hardy's approach. He also doesn't mention AdS/CFT in his last chapter on future developments.

My impression is that Hardy would see AdS/CFT as a coincidence which needs an explanation and that this explanation can't be given from within currently available approaches but only from the yet-to-be-constructed theory of QG which he expects to look very different from string theory.

 

[martinbn]

No, it's supposed to convince people who know something about AdS/CFT that the statement that bulk is encoded in the boundary is a very nontrivial statement.

But you said that you didn't buy it, which suggests more than it being a difficult problem. It suggests that you think it is more likely to be false. Then again in physics almost everything is of that form when it comes to equations, the solution in the bulk is determined by the boundary conditions.

 

[Demystifier]

Then again in physics almost everything is of that form when it comes to equations, the solution in the bulk is determined by the boundary conditions.

Almost everything? Example please!

 

[martinbn]

I cannot think of an example that is not like that! In fact it is a crucial part of what science is! Take any initial value problem, which is what predictability is about, the state of affairs at a given moment say $t=0$ determines the state of affairs at later times. But $t=0$ is the boundary (in space-time) and the later times is the bulk.

 

[Fra]

If one takes an instrumental approach, the only thing that an observer scientifically can say about the "bulk" has to be based on inferences on something that is communicated through the only available communication channel (the boundary).

Thus it seems logically possible that there are hidden information (hidden variables) that are not inferrable from the boundary?

If realism was hard to save with bell experiments, this makes it even harder, as the communication channel constrains what a scientist one one side can ask of the other side. But maybe from Bohmian view, depending on how you see it, it can also be exploited to argue that there is a difference betwen hidden variables such as simply "unknown" or one that are non-inferrable. Bell experiments presumes that there is a define sampling distribution. This does not even make sense for the latter case.

Black Holes, AdS, and CFTs, Donald Marolf
"...Thus, in what some readers may consider an ironic twist, it may be fair to say that in AdS/CFT the Bekenstein-Hawking entropy counts “not the full set of states describing the black hole interior, but only those states which are distinguishable from the outside,” a point of view which has long been championed within the relativity community..."
-- https://arxiv.org/pdf/0810.4886.pdf

/Fredrik

 

[Demystifier]

I cannot think of an example that is not like that! In fact it is a crucial part of what science is! Take any initial value problem, which is what predictability is about, the state of affairs at a given moment say $t=0$ determines the state of affairs at later times. But $t=0$ is the boundary (in space-time) and the later times is the bulk.

Yes, but AdS/CFT is not of that form. In AdS/CFT the boundary does not have a fixed time.

AdS/CFT would be more like something as follows. Consider a sphere $S^2$ as the boundary and the ball inside the sphere as the bulk. Suppose that you want to determine how the ball changes with time. For that purpose you need to specify the Cauchy data at $t=0$ in the whole ball. AdS/CFT would be then like saying that it is actually sufficient to specify the Cauchy data at $t=0$ on the sphere only.

 

[martinbn]

I might be wrong, but I think your analogy is not good. The analogy would be that the boundary is the sphere at all time not just $t=0$ i.e. $S^2\times\mathbb R$ and the bulk is the ball at all times. In other words the boundary is the cylinder and the bulk is the full solid cylinder.

 

[PAllen]

My limited understanding is that ADS/CFT is a statement about theories, not solutions of theories; and that the boundary theory does not include gravity, while the bulk theory does. I would offer the crude analogy of a theory of electric field only on the boundary was equivalent to EM theory in the bulk. This seems altogether different boundary value problems.

 

[Fra]

As I see what it seems on one hand obvious that the boundary complexions (i avoid DOF as that presumes a continuum) in the general case can not encode the information contained in the data in a much larger set. But the adS/CFT kind of duality just conjectures that sometimes this can be done. What does this "coincidence" really mean? I personally see it as a evolutionary tuned setup, and not a conincidence. Most other people might see it as a possible "consistency condition" for observer equivalence.

Ponder that we associated which each observer one "theory", and when these observers are in interacting, they must have a communication channel. This channels constitutes abstractly a common boundary, that obviously typicalyl are of lower dimensionality, right?

Then observer equivalence would required that the two theories (or ALL theories for that matters) are consistent with each other as judged through their common boundary. As no comparasion could bypass this.

But in my view of evolving law, this is not a constraints, its a kind of attractor state, that we expect to be implemented probably right now in the universe, but maybe not in the Planck epoch? or whenever the laws are not yet cooled down.

I have never been a fan of the geometric abstractions, as they tend to encourage you to always make up "mental pictures" and ontologies that are something only confusing.

/Fredrik

 

[martinbn]

My limited understanding is that ADS/CFT is a statement about theories, not solutions of theories; and that the boundary theory does not include gravity, while the bulk theory does. I would offer the crude analogy of a theory of electric field only on the boundary was equivalent to EM theory in the bulk. This seems altogether different boundary value problems.

Well, I don't know what the conjecture states, but I knew it was about theories not solutions to equations. My comment was that "the boundary determines the bulk" is not, at least the phrase, unbelievable. I expect that Demystifier has a deeper reason to expect that the conjecture is not true.

Going back to EM, consider the initial value problem. The two constraint equations have to be satisfied by the initial data, the two evolution equations determine the solution for later times. So, the analogy is that the constraint equations are the equations of a theory on the boundary (the initial hypersurface) and the evolution equations are those of a theory in the bulk. Then there is a one-to-one correspondence between the two theories, given a solution on the boundary there is one in the bulk and the other way around. I know it is not how the AdS/CFT is suppose to be, so I guess unless I learned a bit about it it would not be clear to me why it Demystifier doesn't buy it.

 

[Demystifier]

I might be wrong, but I think your analogy is not good. The analogy would be that the boundary is the sphere at all time not just $t=0$ i.e. $S^2\times\mathbb R$ and the bulk is the ball at all times. In other words the boundary is the cylinder and the bulk is the full solid cylinder.

The AdS/CFT correspondence states that two theories have the same number of degrees of freedom. Loosely speaking, the number of degrees of freedom is the same as the amount of Cauchy data. Since Cauchy data is given at a fixed time (say $t=0$), it should by clear why I think that my analogy is good.

 

[Demystifier]

given a solution on the boundary there is one in the bulk

That's simply not true in classical electrodynamics. There are many different solutions in the bulk that have the same behavior at the boundary. For instance, let $\rho(r)$ be a static spherically symmetric charge density with the property
$$\rho(r)=0 \;\; {\rm for} \;\; r\geq R$$
and a fixed total charge, say
$$\int_0^R \rho(r) 4\pi r^2dr=100$$
(I put $dr$ at the right for you). Different configurations $\rho(r)$ with those properties correspond to different solutions in the bulk that all have the same behavior at the boundary $r=R$.

 

[martinbn]

The AdS/CFT correspondence states that two theories have the same number of degrees of freedom. Loosely speaking, the number of degrees of freedom is the same as the amount of Cauchy data. Since Cauchy data is given at a fixed time (say $t=0$), it should by clear why I think that my analogy is good.

But in your analogy you take a fixed time piece of the boundary and the whole bulk. Why not a fixed time slice of the bulk? That's where the Cauchy data lives anyway. What you describe is not a space and its boundary, but a space and part of a the boundary. I suppose my misunderstanding comes from not knowing what the conjecture is, and until I learn more about it it will not be clear to me, but it is very unlikely for me the be able to find a readable source.

That's simply not true in classical electrodynamics...

No, you are combining the two different analogies. The one in the post to you had nothing to do with EM. It was just an example of a boundary. For that you will need some elliptic equations, otherwise there are plenty of wave examples with fixed boundary (you gave one in the beginning). The EM analogy, in the post to PAllen, has boundary the initial hypersurface.

 

[Auto-Didact]

I'm guessing no one else apart from the OP has of yet read the paper, which is why this thread is getting derailed, starting here:

There is already one good suggestion for quantum gravity - AdS/CFT ...

Hardy's paper is not about AdS/CFT, although he does state in section 17.1 that time evolution of states on spacelike hypersurfaces isn't possible given an indefinite causal structure, and that neither overlapping coordinate patches (or their conformal counterparts), nor gauge fixing can solve this problem.

To get this thread back on track, I'll try to simply illustrate what Hardy is proposing (since I've actually been working on a similar idea myself for different reasons), namely the proposal of a constructivist framework, which is essentially a new conceptual research methodology for theorists for the construction of (physical) theories based on problems with existing theories.

Hardy illustrates this by using a historical example, namely how Einstein tackled the problem of relativistic gravity and how he formulated a physical theory thereof (GR). Hardy distills a few key steps in this reasoning process and generalizes it for his constructive methodology. Somewhat confusingly he calls his constructive methodology an interpretation ("Constructive Interpretation") of QM but it is no such thing as he immediately admits, instead arguing that the interpretative issue of QM will possible resolve itself in some deeper theory.

Summarized, the problem facing Einstein was to find a deeper theory wherein both Newtonian gravity and SR field theories, most prominently Maxwell's electrodynamics, are different limiting cases. Einstein, unlike how most theoretical physicists do today, did this by way of philosophically reasoning about the conceptually conflicting principles underlying the old theories, identifying which are necessary and then through reformulation try to bring them in harmony under one unified conceptual framework consisting of only necessary ingredients. It is only when this step is finished that the mathematics of the theory is modified specifically by replacing the older mathematical formalism with more appropriate mathematics.

This is Hardy's constructive framework:
A. Defining the problem:
Newton Gravity ← Relativistic Gravity → SR Field Theories
B. Philosophical clarification, identification and simplification of the necessary principles and properties:
1. Equivalence principle
2. No global inertial reference frame
3. General coordinates
4. Local physics
5. Laws expressed by field equations
6. Local tensor fields based on tangent space
7. Principle of general covariance
C. Modification of the mathematics of the theory:
I. Prescription: turning SR field equations into GR field equations
II. Addendum: The Einstein field equations
III. Interpretation: geometric interpretation follows naturally from diffeomorphism invariance

This constructive framework is as Hardy says completely general, i.e. it is a theory independent constructive methodology, or more explicitly it doesn't limit itself to any particular theory or formulation of that theory. Instead the framework can, in principle, be used to solve any fundamental problem in physics through the process of analogy. Hardy illustrates this by way of example, i.e. by using the framework to tackle the problem of quantum gravity:

A. Defining the problem:
GR ← Quantum Gravity → SR QFTs
B. Philosophical clarification, identification and simplification of the necessary principles and properties:
1. Dynamical causal structure (from GR) and indefiniteness (from QT)
2. Indefinite causal structure
3. Compositional space
4. Formalism locality
5. Laws given by correspondence map
6. Boundary mediated compositional description
7. Principle of general compositionality
C. Modification of the mathematics of the theory:
I. Prescription: turning QFT calculations into QG calculations
II. Addendum: new physicality conditions for Quantum Gravity
III. Interpretation: will also follow naturally (?)

The particular form of QFT that he utilizes in this example is in his own Operator Tensor QFT formalism (NB: as far as I can see, largely an application of Penrose diagrammatic notation); it goes without saying that this is mathematically equivalent to standard QFT, but the point is:
1) psychologically, it might represent a more natural setting for deriving C.I-III based on the conceptual issues B.1-7
2) in the context of mathematics itself, the correct mathematics needed for an extension to actually carry out C.I and C.II might even already exist.

In any case, I myself am convinced that the adherence to some kind of methodology like this one is necessary to actually make great progress in the practice of theoretical physics today, which has been dominated by overt purely technical reasoning - since the days of Feynman until this very day. Purely technical reasoning has been successful in creating relativistic QFT and the SM, but seems to be hopeless in going beyond them, which is clearly reflected in the now decadeslong stagnation of the field of theoretical physics, where the situation has run amok.

In my opinion, such conceptual frameworks or methodologies, if even partially successful should even be taken a step further, namely not just a framework for one problem, but an entire research programme approaching all fundamental problems. This also shouldn't be done from the single point of view of one theory given some problem, but manifestly opportunistically from the pluralistic point of view of all available competing theories given some problem; this would then enable a direct hierarchical classification and discovery of the interrelationships between (all) physical theories and their possible extensions, in the same spirit as the 8 possible kinematical groups for a uniform and isotropic universe discovered by Bacry and Levy-Leblond.

 

[bhobba]

The whole edifice, loads of calculations and nothing is truly rigorously mathematically justified.

Perhaps you haven't been following Urs's excellent series eg:
https://www.physicsforums.com/insights/newideaofquantumfieldtheory-interactingquantumfields/

As can be seen it can be done in the style you alluded to - the question is does it appeal? Sometimes being non rigorous can be illuminating. The so called zeta function ζ(s) = 1/1^s + 1/2^s + 1/3^s ... has a non-rigorous derivation for s = -k where k is a integer (its used in zeta function regularization and calculation of the Casmir force for instance):

∑(-1)^k*ζ(-k)*x^k/k! = ∑∑ n^k*(-x)^k/k! = ∑ ∑(-nx)^k/k! = ∑e^(-nx). Let S = ∑e^(-nx). e^xS = 1 + S so S = 1/e^x - 1 = 1/x*x/e^x - 1. But one of the definitions of the so called Bernoulli numbers Bk, is x/(e^x - 1) = ∑Bk*x^k/k! or taking the1/x into the sum S = ∑ B(k+1)*x^k/(k+1)! after changing the summation index so you still have powers of x^k. Thus you have ∑(-1)^k*ζ(-k)*x^k/k! = ∑ B(k+1)*x^k/(k+1)!. Equating the coefficients of the power of x^k you have ζ(-k) = (-1)^k*B(k+1)/k+1.

This result implies the bizarre identities ζ(0) = 1+1+1+1... = -1/2 and ζ(1) = 1+2+3+4... = -1/12.

Why such 'silly' results. Well a more rigorous derivation uses contour integration. The infinity you get in summing such things comes from a pole 1/s-1 that appears in the equation when written in a certain form ie where we find ζ(s) - 1/(s-1) is a perfectly well behaved function for all s (you can use the Euler-Maclauren formula to show this if you want - but assume s>1 and use analytic continuation - otherwise the term is an infinite integral). There is a trick using contour integration with what's called a Hankel cut that avoids that infinity - it rears its ugly head at s=1 but its not there otherwise and you get the finite answers.

Rigorous - more difficult - but sound - non-rigorous - not as difficult - but it has issues such as changing the order of the summation in my derivation - why can you do that?

Thanks
Bill