Mathematical Quantum Field Theory - Fields - Comments

[vanhees71]

How can you say that? In your work you don't only discuss S-matrix elements but all the correlation functions, related by Kadanoff-Baym equations!

Yes, but at the end we use these correlation functions to measure spectra of "particles", and these are defined as asymptotic free space. Of course, we do this in the naive mathematically non-rigorous way, using the usual recipies like adiabatic switching and all that. Our conclusion at the time of writing these articles (see, particularly the Annals of Physics one) that one has to do the good old Gell-Mann-Low switching for both "switching on and off the interactions" to make physical sense of the photon spectra. The considered quantities at "finite times" ("transient states") are off by orders of magnitude and, as far as we could figure out, don't have a clear physical interpretation but are calculational tools only.

 

[vanhees71]

And one has to give up the idea that $\hbar$ is a number - instead it is only a formal parameter! And one has to give up the idea that operators act on more than a compact part of space-time - to avoid all the infrared problems.

Well, $\hbar$ is still a number, which is empirically defined. In fact it's a unit-conversion factor and I'm pretty sure that it will be defined officially next year to update the SI for the 21st century and to take legal effect in 2019.

 

[A. Neumaier]

Well, $\hbar$ is still a number, which is empirically defined. In fact it's a unit-conversion factor and I'm pretty sure that it will be defined officially next year to update the SI for the 21st century and to take legal effect in 2019.

But in the theoretical exposition of Urs Schreiber (and implicitly in perturbative QFT in general) it is a parameter in a power series with zero convergence radius. Thus inserting a finite positive value gives results depending on the order of calculation, and diverging if the order is taken too high. Haag and Kastler, to whom he had referred, were using true operators, not formal power series operators.

The considered quantities at "finite times" ("transient states") are off by orders of magnitude and, as far as we could figure out, don't have a clear physical interpretation but are calculational tools only.

Well, at least in the equilibrium case (and in fact more generally in the hydrodynamic limit), they have a very tangible measurable meaning at finite times.

 

[vanhees71]

In the equilibrium case you also have a kind of "adiabatic switching" by pushing the initial time to $-\infty$ to get rid in a subtle way of the necessity to consider the vertical pieces in the extended Schwinger-Keldysh contour. This is another often discussed subtlety in the real-time community. In my opinion it's completely settled in F. Gelis's papers, where it is shown that in fact you can take the initial time finite (but "earlier" than any time argument in the to be evaluted Green's functions), as to be expected from the fact that one deals with equilibrium which is by definition stationary and thus time-translation invariant. From another point of view, it's only important to keep track of the correct "causal regularization" of the on-shell $\delta$ distributions in the free Schwinger-Keldysh-contour propgators, used in perturbation theory.

F. Gelis, The Effect of the vertical part of the path on the real time Feynman rules in finite temperature field theory, Z. Phys. C, 70 (1996), p. 321–331.
http://dx.doi.org/10.1007/s002880050109

F. Gelis, A new approach for the vertical part of the contour in thermal field theories, Phys. Lett. B, 455 (1999), p. 205–212.
http://dx.doi.org/10.1016/S0370-2693(99)00460-8

 

[A. Neumaier]

Right, one uses the positivity of a given state (one of the axioms on a state on an algebra of observables).

In the quoted Nlab article you write (and implicitly you usie this in the present discussion, too):

the definition [of a state] makes sense generally for plain star-algebras, such as for instance for the formal power series algebras that appear in [URL='https://www.physicsforums.com/insights/paqft-idea-references/']perturbative quantum field theory[/URL]

But the notion of positivity is questionable in algebras over rings of formal power series since the latter have no total linear order. Using a partial order instead provides some notion of positivity but not the physical one.In the physical setting, $\hbar_{phys}-\hbar_{formal}\ge 0$, while in the formal setting, this is not the case.

 

[Urs Schreiber]

But the notion of positivity is questionable in algebras over rings of formal power series since the latter have no total linear order.

It's formal power series algebras equipped with star-algebra structure and positivity is defined in terms of the star algebra structure.

 

[strangerep]

I don't understand what's the issue with angular momentum. It's a nice operator algebra of a compact semisimple Lie algebra and as such doesn't make any trouble at all in the standard Hilbert-space theory.

The issue is that I've heard various purist advocates of the algebraic approach suggesting that Hilbert space is not essential to quantum physics, but only an afterthought. I claim Hilbert space is essential, and no one has yet satisfactorily refuted this by deriving the quantum angular momentum spectrum without reliance on Hilbert space.

You construct them algebraically via raising- and lowering operators or, even more convenient, using the fact that the 2D harmonic oscillator has SU(2) symmetry and construct everything with annihilation and creation phonon operators.

Well, the ladder operators come later in a treatment that relies on nothing more than the algebra and abstract Hilbert space. Cf. Ballentine section 7.1. The extra baggage of a harmonic oscillator is unnecessary for deriving the spectrum.

 

[atyy]

One just has to exercise care with regarding this good QM stuff as indication that the Schrödinger picture is a robust foundation for quantum physics in a generality that includes QFT. It turns out that it is not. On the one hand there are intrinsic problems with applying the Hilbert space Schrödinger picture to QFT even in principle (Torre-Varadarajan 98). On the other hand the Heisenberg/interaction picture without a choice of Hilbert space (just with the option to find one, if possible) works wonders and is in fact, more or less secretly, precisely what everyone uses in practice anyway, even if it superficially seems as if Hilbert spaces are being used.

In the Heisenberg picture, there is an initial state which does not evolve with time. The initial state can be any state in the Hilbert space. How can one do away with this arbitrary initial state?

 

[A. Neumaier]

It's formal power series algebras equipped with star-algebra structure and positivity is defined in terms of the star algebra structure.

But as I had mentioned, the positivity obtained is not the physical one, as for formal power series in a variable $x$, the rule $\xi-x\ge 0$ holds for no real $\xi$ while after picking the physical value of $x$ (in a nonperturbative theory) one has $\xi-x\ge 0$ for every real $\xi$ exceeding the physical value.

 

[vanhees71]

Well, the ladder operators come later in a treatment that relies on nothing more than the algebra and abstract Hilbert space. Cf. Ballentine section 7.1. The extra baggage of a harmonic oscillator is unnecessary for deriving the spectrum.

Well, I also think, the Hilbert-space structure is an essential element in teaching at least QM. Since relativistic QFT in (1+3) dimensions is not rigorously defined, I understand that mathematicians try a different approach to define states. Of course, in QT it is of utmost importance to distinguish between observables and states. It's the very point dinstinguishing QT from classical theories that observables and states are disinct entities of the theory.

Concerning the treatment of angular momentum, I never understood, why one should bother students with the wave-mechanical derivation of the angular-momentum eigenvectors, i.e., an old-fashioned treatment of the spherical harmonics. It's so much more transparent to treat the algebra su(2) and its representations. The only cumbersome point is to show that the special case of orbital angular momentum has no half-integer representations, and for that you need the "harmonic-oscillator approach". See, e.g.,

D. M. Kaplan, F. Y. Wu, On the Eigenvalues of Orbital Angular Momentum, Chin. Jour. Phys. 9, 31 (1971).
http://psroc.phys.ntu.edu.tw/cjp/issues.php?vol=9&num=1
http://psroc.phys.ntu.edu.tw/cjp/issues.php?vol=9&num=1
Of course, with that analysis at hand, you can very easily derive all properties of the spherical harmonics by using the position representation (in spherical coordinates).

 

[samalkhaiat]

The issue is that I've heard various purist advocates of the algebraic approach suggesting that Hilbert space is not essential to quantum physics,

This is not correct. Notions such as completeness (by a norm) and continuity (i.e., boundedness) of any element of an operator algebra need to be defined with respect to some vector space topology. Hermitian adjoint can only be defined on a vector space with scalar product. Moreover, every (abstract) non-commutative $C^{*}$-algebra can be realized as (i.e., isomorphic to) a norm-closed , *-closed subalgebra of $\mathcal{L}(\mathcal{H})$, the algebra of bounded operators on some Hilbert space $\mathcal{H}$. Precisely speaking, for every abstract $C^{*}$-algebra $\mathcal{A}$, there exists a Hilbert space $\mathcal{H}$ and injective *-homomorphism $\rho : \mathcal{A} \to \mathcal{L}(\mathcal{H})$. That is $\mathcal{A} \cong \rho (\mathcal{A}) \subset \mathcal{L}(\mathcal{H})$, as every *-homomorphism is continuous (i.e., norm-decreasing).

In general, one can say the following about quantization: Given a locally compact group $G$, its (projective) unitary representation on some Hilbert space $\mbox{(p)Urep}_{\mathcal{H}}(G)$ and the group (Banach) *-algebra $\mathcal{A}(G)$, then you have the following bijective correspondence $$\mbox{(p)URep}_{\mathcal{H}}(G) \leftrightarrow \mbox{Rep}_{\mathcal{H}}\left(\mathcal{A}(G)\right) \ , \ \ \ \ (1)$$ where $\mbox{Rep}_{\mathcal{H}}\left(\mathcal{A}(G)\right)$ is the representation of the (Banach) *-algebra $\mathcal{A}(G)$ on the same Hilbert space $\mathcal{H}$, i.e., *-homomorphism from $\mathcal{A}(G)$ into the algebra of bounded operators $\mathcal{L}(\mathcal{H})$ on $\mathcal{H}$. Similar bijective correspondence exists when $\mathcal{A}$ is a C*-algebra. And both ends of the correspondence lead to quantization. When $G = \mathbb{R}^{2n}$ is the Abelian group of translations on the phase-space $S = T^{*}\left(\mathbb{R}^{n}\right) \cong \mathbb{R}^{2n}$ (or its central extension $H^{(2n+1)}$, the Weyl-Heisenberg group) then (a) the left-hand-side of the correspondence leads (via the Stone-von Neumann theorem) to the so-called Schrodinger representation on $\mathcal{H} = L^{2}(\mathbb{R}^{n})$ [Side remark: of course Weyl did all the work, but mathematicians decided (unjustly) to associate Heisenberg’s name with the group $H^{2n+1}$], while (b) the right-hand-side of the correspondence leads to the Weyl quantization which one can interpret as deformation quantization (in effect, Weyl quantization induces a non-commutative product (star product) on the classical observable algebra, thus deforming the commutative associative algebra of functions $C^{\infty}(\mathbb{R}^{2n})$).

 

[bhobba]

I have said it before and will say it again - I wish Samalkhaiat had the time to post more. His answers cut straight though.

The c*Algebra approach is, as it must be, equivalent to the normal Hilbert-Space approach - but can be formulated in a way where its association with classical mechanics is clearer:
http://www.math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/Gleason.pdf

Thanks
Bill

 

[A. Neumaier]

The c*Algebra approach is, as it must be, equivalent to the normal Hilbert-Space approach

Actually it is more general, as the same algebra may have states corresponding to different Hilbert spaces (more precisely, unitarily inequivalent representations).

Thus it is able to account for superselection rules (restrictions of the superposition principle), which have no natural place in a pure Hilbert space approach.

Also it accounts for quantum systems having no pure states (such as those required in interacting relativistic quantum field theory).