lim
L→∞
1 +e−βHΛL 1−e−βHΛLf, f
(6.241)
≤ 1
1−m2ν2
1 +e−βH 1−e−βHf, f
.
Combining all of these inequalities, we have, for anym∈Nν, the inequalities 1 +e−βH
1−e−βHf, f
≤lim inf
L→∞ ρ(m)ω,Λ
L(β, zL)≤lim sup
L→∞
ρ(m)ω,Λ
L(β, zL)≤ 1 1−m2ν2
1 +e−βH 1−e−βHf, f
. (6.242) Applying these bounds to eq. (6.232), and lettingm→ ∞, we have
lim
L→∞
X
k∈Nν,k6=1
1 +zLe−β(Lπk)2 Lν(1−zLe−β(πLk)2)
Lν Z
Rν
dνx f(x)ψk,L(x)
2
=
1 +e−βH 1−e−βHf, f
. (6.243) We have thus shown that
L→∞lim
1 +zLe−βHΛL 1−zLe−βHΛLf, f
= 2ν(ρ−ρc) Z
Rν
dνx f(x)
2
+
1 +e−βH 1−e−βHf, f
. (6.244) Finally, if we now consider the form ofωΛL from proposition 6.3, we have shown that
lim
L→∞ωΛL(W(f)) =ω(W(f)), (6.245)
andω has the desired value forW(f).
7 Conclusion and Further Discussion
7.1 Bose-Einstein Condensation
The ultimate result of the previous section proposition 6.6 states that with a very specific process of taking the thermodynamic limit, namely, we must vary the activity to account for a change in the volume so as to maintain a fixed density, one is able to construct a state which presents two distinct phases of matter depending on the fixed density.
The first state, which occurs if we are under the critical density, is clearly just a regular non-interacting gas. The second state is the condensate. By comparison of theρωfunction in propo-sition 6.6, we see that above the critical density there is the addition of a scalar term. The effect of this additional scalar term is that physically one tries to add additional particles to the system, which corresponds to increasing the fixed densityρ, instead of distributing ”evenly”, in some sense, they will immediately go to the lowest energy state and increase this scalar term.
Mathematically, we have shown that by using Gibbs grand canonical equilibrium states, and an appropriate limiting procedure, there exists a well-defined state on a physically relevant algebra of operators which shows a distinct splitting of the regular gas and condensate phases.
7.2 Mathematical Content of this Thesis
The first half of the thesis was used to construct and show that there is a rather broad and rich algebraic theory surrounding the field of quantum statistical mechanics. In a sense, one can con-sider the first half of this thesis to be a light introduction to certain field-theoretic arguments and
methods.
The main result of the first half of the algebraic part of this thesis is theorem 4.2. By virtue of being a uniqueness theorem, the theorem roughly states that for certain algebraic and topolog-ical properties, it is enough to study any C∗-algebra which is generated by the Weyl operators.
Going further, if one wishes to extract abstract versions of the annihilation and creation operators, with the help of additional regularity conditions in the form of analyticity and strong continuity of the represented forms of the time evolutions given by the Weyl operators, then we are able to do so as shown in section 5.2.
One of the interesting facts in the first half of the thesis concerned the analytic states in sec-tion 5.2. One will recall that the analytic states were regular states on someCCR-algebra which satisfied additional regularity properties in the form of analyticity of the state acting on a gener-ator. Remarkably, these conditions were sufficient to yield a Hilbert space, a representation and an analytic vacuum vector of the state.
One can specify the commutation relations in the form of Weyl operators, then consider a state on the algebra which has some physical significance, and hope that this state is analytic. In this way one avoids talking about the concrete Hilbert space and vacuum vector, and, they are instead given for free. The issue here is that the construction of the Hilbert space and vacuum vector are a part of the GNS construction. In practice, one could try and explicitly compute what the GNS construction yields and identify the Hilbert space and vacuum vector with something concrete. Of course, in this case, one would presumably also already be able to approach the problem without the GNS construction.
The latter half of the thesis could actually be considered an example of this sort of approach.
In the latter half, we did an explicit construction and extension of the Gibbs equilibrium state to include the annihilation and creation operators. In doing so, through various computations, we were able to calculate the value of the Weyl operator for rather general Hilbert space. To specialize to taking the thermodynamic limit, we of course had to include such concepts as volume which show up in the free particle Hamiltonian in a open and bounded set. Ultimately, we constructed the state corresponding to the two-phase gas, in which one of the phases was the condensate.
Having constructed these states, one might be tempted to try and generalize the abstract prop-erties of these states rather than trying to come up with a new Hilbert space and Hamiltonian to study. We will discuss this idea in the next subsection.
The previous paragraphs give a succinct, but shallow, summary of the second half of the the-sis. One notes that the first mathematical issues that one faces is the extension of the Gibbs state to the polynomials of annihilation and creation operators. This problem is dealt with by showing thate−βKµ, along with some additional properties, is sufficiently ”small”, in some operator norm sense, so as to counteract the unboundedness of the creation and annihilation operators. This informal statement is made rigorous in section 6.2.
Next, one must contend with doing calculations inside the trace without the help of bounded operators. To this end, we make frequent applications of the method of computation used in eq. (6.60). For an example of the usage of this method, we suggest filling in some of the details of eq. (6.82). Finally, one computes the value of the Weyl operators using methods from [16], and shows convergence of states via various technical lemmas.
We remark that the mathematics used in the computations starting from section 6.5, barring some of the more technical results such as lemma 6.3 or the strong-graph limit theorem used in proposition 6.3, is relatively standard and basic. For the most part, we are simply computing upper and lower bounds, and using basic analysis.
7.3 Further Work
The ultimate constructed state in proposition 6.6 of this thesis shows a clear and physical splitting of phases. A natural point from where to generalize is to consider states in which there is more than a single scalar addition to the density. For instance, in this example of the condensate, there is a single energy level to which the particles all fall to. But what is stopping us from having a finite convex combination of other possible ”ground states” in the condensate? In fact, it is trivial to see that one can write an analytic state which implements this precise idea.
It is less trivial to see how one would arrive at this state without just explicitly giving the value of the Weyl operators. In a sense, we would like to write something of a generalization to the single species condensate in which there is only a single ground state to which the particles flow to.
Another direction one could take is to consider different Hamiltonians. For instance, one can consider the ultrarelativistic gas in which the free particle Hamiltonian is given byHΛL =pΛL. There are certain relevant theorems such as the theorem concerning strong graph limit in propo-sition 6.3 which initially seem to be applicable to this case as well. The biggest possible problem are the non-trivial theorems concerning the path integrals in lemma 6.3. Nevertheless, one can take any relevant Hamiltonian and use the ideas of the thermodynamic limit and some of the computations as a guide to how to approach this problem with another Hamiltonian.
The mathematical possibilities with the algebraic approach are endless. One can casually pe-ruse the first section of [1] and find numerous interesting and relevant topics to study and clarify.
To be more specific, for readers interested in studying the algebraic side of quantum statistical me-chanics, we suggest first gaining an understanding ofW∗-algebras and semi-group theory from [2].
One can then study the algebraic results which were left out from this thesis in [1, 5.2. Contin-uous Quantum Systems I]. A natural continuation of this thesis would be the next section of [1]
which focuses on KMS-states. In fact, this thesis used a form of the KMS-condition given in eq. (6.5). However, the approach used in [1, 5.3. KMS States] is considerable more abstract and begins withC∗-dynamical systems.
For readers interested in studying the physics of quantum statistical mechanics, we recommend the classic text [8]. For something related to quantum statistical mechanics, one could study the quantum Bose liquid and its relation to superfluidity in [8][Part 2, Section 3]. For a modern, and thoroughly advanced, book on the kinetic theory of Bose-condensed gases, we suggest [5].
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