Tag: Analysis

Over the past few months, I have been trying to understand the so-called Sine-Gordon transformation, so I have posted a few questions on this topic here. I also did extensive research on this subject, so I came to some conclusions. Still, I have a few questions that I would like to share with you. I will first draw the general picture and some of my conclusions.

We consider a function $ V: mathbb {R} ^ {n} times mathbb {R} ^ {n} $ who is continuously differentiable, satisfied $ sup_ {x, y in mathbb {R} ^ {n}} | V (x, y) | le K $ and
$$ langle f, Vg rangle: = int _ { mathbb {R} ^ {n}} int _ { mathbb {R} ^ {n}} f (x) V (x, y) g (y) ddy ge 0 tag {1} $$
for each $ f, g in L ^ {2} ( mathbb {R} ^ {n}) $. If we define $ B: mathcal {S} ( mathbb {R} ^ {n}) times mathcal {S} ( mathbb {R} ^ {n}) $ to be $ B (f, g) equiv langle f, Vg rangle $, the associated quadratic form $ f mapsto B (f, f) $ is non-negative, so that, according to Minlos' theorem, there is a (Gaussian) measure $ mu_ {V} $ sure $ mathcal {S} & # 39; ( mathbb {R} ^ {n}) $ such as
$$ W (f): = e ^ {- frac {1} {2} B (f, f)} = int _ { mathcal {S} & # 39; ( mathbb {R} ^ {n })} d mu_ {V} (T) e ^ {iT (f)} $$
Because $ mathcal {S} ( mathbb {R} ^ {n}) subset mathcal {S} & # 39; ( mathbb {R} ^ {n}) $, each $ f in mathcal {S} ( mathbb {R} ^ {n}) $ induces a distribution in $ mathcal {S} & # 39; ( mathbb {R} ^ {n}) $. So if we fix it $ epsilon_ {1}, …, epsilon_ {N} in mathbb {R} $ and $ x_ {1}, …, x_ {N} in mathbb {R} ^ {n} $, we can choose sequences $ {f_ {l} ^ {(j)} } _ {l in mathbb {N}} $ such as $ f_ {l} ^ {(j)} to epsilon_ {j} delta_ {x_ {j}} $, for each $ j = 1, …, N $. We can prove that:
$$ lim_ {l to infty} int _ { mathcal {S} & # 39; ( mathbb {R} ^ {n})} d mu_ {V} (T) prod_ {j = 1} ^ {{N}: e ^ {iT (f_ {l} ^ {(j)})}: _ {V} = e ^ {- sum_ {1 le i <j le N} epsilon_ {i} epsilon_ {j} V (x_ {i}, x_ {j})} $$
or $: e ^ {iT (f)}: _ {V}: = e ^ {iT (f)} e ^ { frac {1} {2} B (f, f)} $. Let's introduce the notation:
begin {eqnarray}
lim_ {l to infty} int _ { mathcal {S} & # 39; ( mathbb {R} ^ {n})} d mu_ {V} (T) prod_ {j = 1} ^ {N}: e ^ {iT (f_ {l} ^ {(j)})}: _ {V} equiv bigg { langle} prod_ {j = 1} ^ {N}: e ^ {i epsilon_ {j} T (x_ {j})}: _ V bigg { rangle} _ {V} tag {2} label {2}
end {eqnarray}
The right side of ( ref {2}) makes no sense once $ T $ cannot be assessed on an ad hoc basis. However, this is just a notation. Now the partition function of a system in the grand canonical ensemble is given by:
begin {eqnarray}
Xi _ { Lambda} ( beta, z) = 1+ sum_ {N = 1} ^ { infty} frac {z ^ {N}} {N! 2 ^ {N}} sum _ { substack { epsilon_ {j} = pm 1 \ j = 1, …, N}} int _ { Lambda ^ {N}} dx_ {1} cdots dx_ {N} e ^ {- beta sum_ {1 le i <j le N} epsilon_ {i} epsilon_ {j} V (x_ {i}, x_ {j})} tag {3} label {3}
end {eqnarray}
So, we can rewrite ( ref {3}) using the notation in ( ref {2}):
begin {eqnarray}
Xi _ { Lambda} ( beta, z) = 1+ sum_ {N = 1} ^ { infty} frac {z ^ {N}} {N! 2 ^ {N}} sum _ { substack { epsilon_ {j} = pm 1 \ j = 1, …, N}} int _ { Lambda ^ {N}} dx_ {1} cdots dx_ {N} bigg { langle} prod_ {j = 1} ^ {N}: e ^ {i sqrt { beta} epsilon_ {j} T (x_ {j})}: _ V bigg { rangle} _ {V} tag {4} label {4}
end {eqnarray}
This motivates another simplification of the notation. We interpret the integrand in ( ref {4}) as a $ N $ iterated integrals, in order to write:
begin {eqnarray}
frac {1} {2 ^ {N}} sum _ { substack { epsilon_ {j} = pm 1 \ j = 1, …, N}} int _ { Lambda ^ {N }} dx_ {1} cdots dx_ {N} bigg { langle} prod_ {j = 1} ^ {N}: e ^ {i sqrt { beta} epsilon_ {j} T (x_ {j })}: _ V bigg { rangle} _ {V} equiv bigg {(} frac {1} {2} bigg { langle} sum _ { epsilon = pm 1} int _ { Lambda} dx: e ^ {i sqrt { beta} epsilon T (x)}: _ {V} bigg { rangle} _ {V} bigg {)} ^ {N} equiv bigg {(} bigg { langle} int _ { Lambda}: cos sqrt { beta} T (x): _ {V} dx bigg { rangle} _ {V} bigg { )} ^ {N}: = langle C _ { Lambda, beta} rangle_ {V} ^ {N} tag {5} label {5}
end {eqnarray}
Finally, we have:
begin {eqnarray}
Xi _ { Lambda} ( beta, z) = sum_ {N = 0} ^ { infty} frac {z ^ {N}} {N!} Langle C _ { Lambda, beta} rangle_ {V} ^ {N} equiv langle exp (z C _ { Lambda, beta}) rangle_ {V} tag {6} label {6}
end {eqnarray}
The relation ( ref {6}) is called the Sine-Gordon transformation. Having said that, I would like to raise a few questions.

(1) If my reasoning is correct, the representation of Sine-Gordon is formal, in the sense that it does not represent a proper Gaussian integral; instead, it's just a matter of scoring. If that is the case, I agree with that, but I don't understand the point here. If ( ref {6}) is just a scoring question, why is it useful? If I draw a conclusion with ( ref {6}) as a starting point, why should it really hold if all of this is formally written? I know that Gaussian integrals are useful tools but it is not an appropriate Gaussian integral, right?

(2) Is it possible to give a precise meaning to ( ref {6})? Is there a construction where $ Xi _ { Lambda} ( beta, z) $ is a real Gaussian measurement and, if so, how to do it? (It is not unusual to come across an article where the Sine-Gordon transformation is treated as a true mathematically significant representation, so I wonder if I am not reading it correctly or if there is in fact a significant version).

(3) In practice, the notation ( ref {2}) is useful because it allows us to perform certain formal operations such as the exchange of integrals and products as in ( ref {5}) and ( ref {6}). Can any of these operations be properly justified? In other words, to what extent ( ref {6}) does it hold only as a formal series?

Note: To be complete, I based this post mainly on the work of Fröhlich and the work of Fröhlich and Park. Other good references are the work of Brydges and Federbush and the work of Dimock.