Please note that I do not have a background in physics, so if possible please refrain from a bunch of $ |x\rangle $ notations, unless clearly specifying what it symbolically means.

So I have been learning about representation theory lately in particular I have studied square integrable irreducible representations, and I'm interested in the applications of these. I have come to understand that given a square integrable irreducible representation $ U $ of a locally compact group $ G $ on a Hilbert space $ \mathcal{H} $ and an admissible vector $ g \in \mathcal{H} $, then the orbit $ \mathscr{O}_g := \{U(x)g\mid x\in G\} $ is a coherent state. Furthermore if the group $ G $ is the (Weyl-)Heisenberg-group, then these coherent states are "classical coherent states"(?).

So I understand from this that coherent states can be described by these collections of vectors/functions in a Hilbert space and sometimes they constitute frames and possible wavelets(?). How exactly is such a collection of vectors understood in the context of coherent states? What does a coherent state describe? and why are they interesting?

If you can refer me to articles or literature explaining these questions in terms understandable by someone who has mostly had basic mechanics then it'd be much appreciated.

  • $\begingroup$ I suppose you are familiar with the classical example of coherence , soldiers breaking step when crossing a bridge? en.wikipedia.org/wiki/Broughton_Suspension_Bridge . Coherence in physics means being in step, and when waves are in the discussion they are coherent if the phases are fixed cf lasers, coherent light paragraph hyperphysics.phy-astr.gsu.edu/hbase/optmod/qualig.html $\endgroup$
    – anna v
    Jun 13 '14 at 11:50
  • $\begingroup$ So coherence is when something is in sync, for instance oscillations? What I would really like though, is a bit of insight into how these coherent states (as in collections of vectors) are applied in physics. $\endgroup$
    – zo0x
    Jun 13 '14 at 12:40
  • $\begingroup$ I can give you an example of a coherent state in quantum mechanics. The energy eigenstates wavefunctions of the harmonic oscillator have significant amplitude over the entire "classically allowed" region of space. It's hard to see how this relates to a ball and spring, in which the ball exists in a definite location, and oscillates. A state can be constructed (the coherent state) where the wavefunction is localized at the classical location, and it does oscillate. It's also the minimum uncertainty wavefunction. (I can't help with your math, nor a classical example.) $\endgroup$
    – garyp
    Jun 13 '14 at 12:49
  • $\begingroup$ I gave you an example with the lasing . Each photon is represented by one of your vectors, and they are coherent if they have a fixed phase when represented as a sinusoidal function. read the previous paragraphs $\endgroup$
    – anna v
    Jun 13 '14 at 13:04

I will define coherent states in the context of Fock spaces (I think it's simpler than to define them in quantum mechanics, and historically more accurate; for the q.m. ones see the reference in the end). Given any separable Hilbert space $\mathscr{H}$, we can define the symmetric Fock space $\Gamma_s(\mathscr{H})$ as: $$\Gamma_s(\mathscr{H})=\bigoplus_{n=0}^\infty \mathscr{H}^{\otimes_s n}$$ where $\mathscr{H}^{\otimes_s 0}=\mathbb{C}$ and $\otimes_s$ is the symmetric tensor product. The antisymmetric Fock space is the same substituting symmetric products with antisymmetric ones. I will focus on the symmetric situation, even if coherent states may be defined also for antisymmetric Fock spaces (with the aid of Grassmann algebras).

On $\Gamma_s(\mathscr{H})$ the basic (unbounded) operators are the creation and annihilation operators $a^*(f)$ and $a(f)$, $f\in \mathscr{H}$. They are one adjoint of the other, and are the closure of $$a(f)g^{\otimes n}=\sqrt{n} \; \langle f , g\rangle_{\mathscr{K}} \; g^{\otimes (n-1)}$$ $$a^{*}(f)g^{\otimes n}=\sqrt{n+1} \; f\otimes_s g^{\otimes n}$$ You can find a good reference, domain of definitions and other information on the second volume of the books by Reed and Simon "Methods of modern mathematical physics"(section on free quantum fields). You can then define the Weyl operators $$W(f)=\exp\{i(a^*(f)+a(f))\}\; ,\; f\in \mathscr{H}\; .$$ They are unitary and satisfy the Weyl relations $$W(f)W(g)=W(f+g)e^{-i\Im \langle f,g\rangle_{\mathscr{H}}}\; .$$ Additionally, they translate the creation and annihilation operators, and have a lot of other nice properties. The coherent states are defined as $$W(f)\Omega\; ,$$ where $\Omega$ is the Fock space vacuum, i.e. the state with only nonzero component the unit vector of $\mathscr{H}^{\otimes_s 0}=\mathbb{C}$.

The Weyl relations are closely related to representation theory (even if I don't know much about that), and that may be the link with the representations of your Weyl-Heisenberg group.

These vectors are very relevant in a lot of aspects of physics, e.g. in semiclassical analysis. From an experimental point of view, they are easy to prepare, especially when you have to deal with radiation (quantum optics).

An exhaustive and recent mathematical review on coherent states may be found in this book.

  • $\begingroup$ Why coherent states $W(f)\Omega$ are equal to $e^{-\frac{1}{2}||f||^{2}_{\mathcal{K}}} \sum_{n=0}^{\infty} \frac{f^{\otimes n}}{\sqrt{n!}}$ ? $\endgroup$ Feb 5 at 0:23
  • $\begingroup$ @GabrielPalau It follows from a direct computation, writing the exponential as a series (this is possible on the vacuum, for the latter is an analytic vector for the creation and annihilation operators). $\endgroup$
    – yuggib
    Feb 5 at 8:27
  • $\begingroup$ And probably, if I recall correctly, the formula holds for $W(\frac{f}{i})\Omega$ rather than $W(f)\Omega$. $\endgroup$
    – yuggib
    Feb 5 at 8:38

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