Timeline for Two point function of a free scalar field in Euclidean space-time
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10 events
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| May 3, 2016 at 3:44 | comment | added | asv | @user1504: You are right. However my argument shows in the same way that the function $\langle 0|\partial\phi(x)\partial\phi(0)|0\rangle$ is harmonic and hence smooth everywhere. This is already suspicious since at $x=0$ I would expect singularity. | |
| May 2, 2016 at 19:52 | comment | added | user1504 | If you watch di Francesco et al carefully (instead of listening to what they say), you will see that they never actually use the 'operator' $\phi(x)$ as anything but a mnemonic. It is not used to construct states from the vaccum; instead one uses it to construct $\partial^n \phi(x)$ and $e^{ik\phi(x)}$ and then uses these operators to construct states. The symbol '$\phi(x)$' is not an operator on any of these Hilbert spaces. | |
| Apr 30, 2016 at 12:52 | comment | added | asv | Many thanks for your comments. In that case I would like to ask my original question in 2d: how to compute $D(x)$? Is $D(x)$ constant function? Apparently the argument of my post does work (provided one replaces $1/|x|^{D-2}$ with $\log|x|$); the exception is the remark at the end of my post which should be ignored in 2d. | |
| Apr 30, 2016 at 12:36 | comment | added | Igor Khavkine | @sva, in 2D a special trick is possible because of the D'Alembert formula for solutions of the wave equation $\square \phi(t,x) = 0$, $\phi(t,x) = \phi_+(x-t) + \phi_-(x+t)$. This makes it possible to analytically continue $\phi_+$ and $\phi_-$ as respectively holomorphic and anti-homolorphic functions of $z=x+it$. See this older answer for a bit more detail. | |
| Apr 30, 2016 at 8:03 | comment | added | asv | Does that mean that for $D>2$ there is no operator formalism while for $D=2$ it does exist? In that case I would like to include the case $D=2$ to my original question. | |
| Apr 30, 2016 at 7:56 | comment | added | asv | The main reason why I excluded the case $D=2$ from my post is that even the 2d Minkowski space-time is quite special: in quantization (due to Wightman) of massless scalar field in 2d Minkowski space-time one has to abandon the requirement of positive definiteness of the scalar product, and there is some extra-parameter which might be arbitrary. For a partial information see the answer to this post: physics.stackexchange.com/questions/246315/… But this seems to be irrelevant for my argument here. | |
| Apr 30, 2016 at 7:50 | comment | added | asv | Now I am quite puzzled by this answer. Almost the same argument I gave works well in 2d space-time (one has to replace $\frac{1}{|x|^{D-2}}$ with $\log|x|$). However in the literature on CFT one explicitly uses the operator formalism, see e.g. the whole Ch. 6 of "CFT" by Di Francesco, Mathieu, Senechal which is dedicated to it. On p.150 it appears: "Hilbert spaces and operators are nonetheless extremely useful in CFT...". The free scalar massless field seems to be the simplest example of CFT; it is considered in $\S $6.3. | |
| Apr 29, 2016 at 19:01 | comment | added | Igor Khavkine | Yes, that's essentially right. There is no problem with the path integral formalism, because it connects to the operator formalism via time ordered vacuum expectation values, where we already know that analytic continuation works: $\langle 0 | T(F[\phi]) | 0 \rangle = \int D\phi \, F[\phi] e^{iS[\phi]}$, where on the right $F[\phi]$ is a scalar valued (possibly non-linear) functional, while on the left $T(F[\phi])$ is the operator form of its time-ordered quantization. | |
| Apr 29, 2016 at 17:52 | comment | added | asv | Just to make sure: do I understand correctly that you claim that no operator version of Euclidean free scalar massless field exists, as opposed to its Minkowski version? If this is the case, in what version it does exist at least on the physical level of rigor (e.g. path integrals formalism)? | |
| Apr 29, 2016 at 17:33 | history | answered | Igor Khavkine | CC BY-SA 3.0 |