On Monday, I will be at HU Berlin to give a seminar on
my loop cosmology paper (at 2pm in case you are interested and around). Preparing for that I came up with an even more elementary derivation of the polymer Hilbert space (without need to mention C*-algebras, the GNS-construction etc). Here it goes:
Let us do quantum mechanics on the line. That is, the operators we care about are
![x](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sDlGnqF3SBi11RxxWz7OVnE_sbsUnN54AfKdenNHFU7Mp3mmGHN3DhMGKn5wHuO_xm8Q4KCM-9bQKYLCKhUsUPMqQMDMY1R0sT9PnsRCDOtsAkTA=s0-d)
and
![p](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uJIEc563cadzGfEL9f_soRts90lqstFT93-E6uWAUNpj2ksrv9_mAXdu0M3Ck2S_gMJFZsA_RWN5w6tHiXf7bdODpGuL6-b-mTexo5m33l9_3tPQ=s0-d)
. But as you probably know, those (more precisely, operators with the commutation relation
![[x,p]=i](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_ttYb_phOMuPErYGnyWavTCYF1FOn8m30HHDx1Ey7q5RXkeRjaweNPmEj6w7tTgx6rOvRwI3L9UlP0KOGgdpMajSHUNct7FTxU46rlENNlncj8XoPWjDrCaIIytOVxI=s0-d)
) cannot be both bounded. Thus there problems of domains of definition and limits. One of the (well accepted) ways to get around this is to instead work with Weyl operators
![U(a)=\exp(iax)](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_ttYpYdvO-51U4q65q1YjPYn4aR_6lm2SyuzQKRm06iWpLu39rfcC4WN4uiXC3RXVPFiRMAYJ6F0squDGTO9Cu5bCS2g3uAlf-SMaQVJ5ny0W6x2NpnlCta1xYN7fBS4trb-Fk=s0-d)
and
![V(b)=\exp(ibp)](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vMHsetVARDsx7hWn_Qj3KW-ie13H1yuIhkPAaaqu8QvtB9HDoIx3vYo2PJKagj5WqOWq_q2-HK8MUXllEOCZD7ZYwLDNyFf7RmkEkv_PWpRaNFI1v4wprB36fA-nJW_KUUYcc=s0-d)
. As those will be unitary, they have norm 1 and the canonical commutation relations read (with the help of B, C and H)
![U(a)V(b)=V(b)U(a)e^{iab}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_t75WeCDLx-RlZm7oX4QuaXttnEvXxKisV7SGTraXiyowvD5rrH6JAkgeM7eCNkFr5d6RfLhDtFVvUpU23Kib1kRpS8PptQm_2xEoTvmHVXRhd3DSMD9BICMBO_PWn1Y4dFyJ7n9jiHh7TxHlI619P9mA=s0-d)
. If you later want, you can go back to
![x=dU(a)/da|_{a=0}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_s5NXR1mLb-44w7NH6SsoUZGMAXvmmwMJISZbHdeowN4d1iqkCAJ2TeNxp-MFYy5YCyNfBvhUt_s9RoxGvWwdF-OEG6VGnEFcFokdv1YTrNooIkxGpWKwFhqJaOEeDn9WN8d7abK1ulKnDZV4A=s0-d)
and similar for
![p](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uJIEc563cadzGfEL9f_soRts90lqstFT93-E6uWAUNpj2ksrv9_mAXdu0M3Ck2S_gMJFZsA_RWN5w6tHiXf7bdODpGuL6-b-mTexo5m33l9_3tPQ=s0-d)
.
Our goal is to come up with a Hilbert space where these operators act. In addition, we want to define a scalar product on that space such that
![U](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vV0vrYXvtR-ZEQvA4KW4LbyOoeE5-s-WxeYjfVxSjSJDQK0yQ8YiVdMnz3CXGJSw463uxpKWIIlARW8Apem7WiApgITavwb4cu9X_eSwvGJLgTxZI=s0-d)
and
![V](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sUPtxH42_7kmOTnwnsbdDKS17cZBz_nNtqWgyQU7JhHWe2V-1beM_8U3Q4_tHXFlVaE6UJeD4ZvsWRwkNtBSLnE_J3JKdDJDdOq76_gOya90iWtW0=s0-d)
act as unitary operators preserving this scalar product. We will deal with the position representation, that is wave functions
![\psi(x)](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_t5aKkbWU-haBrZxfV_yfhRWzS1F9h65qfmRBmC6orTR-gJhgm-xl5A7YKykp2ha09pcgyCHqilLDmYlZpPWxjIBDxSRYi2jE1JeE_tY59brotL7sJm88fc88NJcg=s0-d)
.
![U](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vV0vrYXvtR-ZEQvA4KW4LbyOoeE5-s-WxeYjfVxSjSJDQK0yQ8YiVdMnz3CXGJSw463uxpKWIIlARW8Apem7WiApgITavwb4cu9X_eSwvGJLgTxZI=s0-d)
and
![V](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sUPtxH42_7kmOTnwnsbdDKS17cZBz_nNtqWgyQU7JhHWe2V-1beM_8U3Q4_tHXFlVaE6UJeD4ZvsWRwkNtBSLnE_J3JKdDJDdOq76_gOya90iWtW0=s0-d)
then act in the usual way,
![V(b)](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sTgLKjgVy-p1BEoZyKs9_jiDSr1Cdd629q7O3rNAo5tdS4vzKyWUb0DGE0GUztPiZEk8d67KXCH1i0hXvkOKcDQJwjT7sG4_syiRpiRwlhOukAwcgjgp0=s0-d)
by translation
![(V(b)\psi)(x)=\psi(x-b)](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_ub54vpx9Y2x9mnK6YRmP6XIec0BO4HIMQ6A1IW77HgtDbXpJH6dLBZyITFIl8N5tCcnG4TQBDGEv_46QA1XQujkg6MH1PL7t2zWalaBijItJnEUutoRVvRQtqTQsArxFZ2hK7G0ldvrJa3th2A=s0-d)
and
![U(a)](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_ubaSe0ccVrpqIJkDtkOjZiH4-E-3Yc4DEk292qls3x15dTutxWhk19Pzs_vN62pvEIEmGNM_SKqHGMqrPbEAWXXM4BJQWoJqEho32F3nlPFauycl3_0pE=s0-d)
by multiplication
![(U(a)\psi)(x)=e^{iax}\psi(x)](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_v3xEW9w0ZAtx9hyKMCXaqVlq-AYDtswoTv7KRaecqDNIEJeDgsRB6bQhp3gl_SwieDuBYrgfUFAM6Lz9HJRdBNYBVjMm1pJ0I1E3qSWeua4vNZHKg9j5xXufYNj38BKW9G6B48nhK--m0rN8_yg-34ScP09gbvApKH=s0-d)
. Obviously, these fulfil the commutation relation. You can think of
![U](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vV0vrYXvtR-ZEQvA4KW4LbyOoeE5-s-WxeYjfVxSjSJDQK0yQ8YiVdMnz3CXGJSw463uxpKWIIlARW8Apem7WiApgITavwb4cu9X_eSwvGJLgTxZI=s0-d)
and
![V](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sUPtxH42_7kmOTnwnsbdDKS17cZBz_nNtqWgyQU7JhHWe2V-1beM_8U3Q4_tHXFlVaE6UJeD4ZvsWRwkNtBSLnE_J3JKdDJDdOq76_gOya90iWtW0=s0-d)
as the group elements of the Heisenberg group while
![x](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sDlGnqF3SBi11RxxWz7OVnE_sbsUnN54AfKdenNHFU7Mp3mmGHN3DhMGKn5wHuO_xm8Q4KCM-9bQKYLCKhUsUPMqQMDMY1R0sT9PnsRCDOtsAkTA=s0-d)
and
![p](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uJIEc563cadzGfEL9f_soRts90lqstFT93-E6uWAUNpj2ksrv9_mAXdu0M3Ck2S_gMJFZsA_RWN5w6tHiXf7bdODpGuL6-b-mTexo5m33l9_3tPQ=s0-d)
are in the Lie algebra.
Here now comes the only deviation from the usual path (all the rest then follows): We argue (motivated by similar arguments in the loopy context) that since motion on the real line is invariant under translation (at least until we specify a Hamiltonian) is invariant under translations, we should have a state in the Hilbert space which has this symmetry. Thus we declare the constant wave function
![|1\rangle=\psi(x)=1](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vupHn0DHz_Yg9FyYulpd3GaRz6TrKJ1SnUh74D86XsnJTMMWTCuOmzd_3lZyUvks6Z_5y2Wnr-CQHIeTlcoHAT6dmx6tQrf6vCtnIlVkqhGZEplDf-eImR7N_xreFt_3wk9q1hU-nzekAnEyK0qg=s0-d)
to be an element of the Hilbert space and we can assume that it is normalised, i.e.
![\langle 1|1\rangle=1](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vBUeJqgVYmDLLbTe-Zb2cbFjCJqR7br5RLxIJ4zuPcTOeZdeOX_0N7pNlKtUyw0_up3LNoysppTcBzSxdaalAs1dqCGEvjaPKXgICgl4WVWSoAAwmTiMiVEAOp8kU9bGjHtunMvQAlBdhprqY7=s0-d)
.
Acting now with
![U(a)](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_ubaSe0ccVrpqIJkDtkOjZiH4-E-3Yc4DEk292qls3x15dTutxWhk19Pzs_vN62pvEIEmGNM_SKqHGMqrPbEAWXXM4BJQWoJqEho32F3nlPFauycl3_0pE=s0-d)
, we find that linear combinations of plane waves
![e^{ikx}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_usXIIhFgb715mDrNJbKFTclxc5e-uf5xLyVBuCOn2qLtFda17YXydKXYcqjs8imDk-U8EhaKpK6IK2GS-9FW9_f0RqvUTc8htOf21d6pQz0VqUgx5VJCgLNBvrtXyiGys=s0-d)
are then as well in the Hilbert space. By unitarity of
![U(a)](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_ubaSe0ccVrpqIJkDtkOjZiH4-E-3Yc4DEk292qls3x15dTutxWhk19Pzs_vN62pvEIEmGNM_SKqHGMqrPbEAWXXM4BJQWoJqEho32F3nlPFauycl3_0pE=s0-d)
, it follows that
![\langle e^{ikx}| e^{ikx}\rangle =1](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uBgofZzYRQ2-mVO9L8Gk3B5PUWpZsz_EJkQetXQHfZeWi9mfo9lSuubrc29qx6WYUN77MuYAZq9X2JC_HHTlrTmPV02mBIF6IrmsKs1Ssk9D9WTkHlwJrOqQFQ9yEXTpHyQU9P3fih4Rf_UFhxlQYgWwTFzZAPsClcPAY1-9aCH-cJzHIwKjg=s0-d)
, too. It remains to determine the scalar product of two different plane waves
![\langle e^{ikx}|e^{ilx}\rangle](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vq5FzGwgwB6GjdrZMkPT-34rz-kJu03uEk8O53ZFhZZ4fS83cmcGf8HMZ4nJvC0L1DDrMYifwdaWV1vMT7pIAW7RSUtBcWBCSMhfoz5kwtBdltbt10MdlrYCBiRDstWWYeWNaL_xTvqwe0fBDcKjQraopisdojSpIZbYV4x9rugYx_eg=s0-d)
. This is found using the unitarity of
![V](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sUPtxH42_7kmOTnwnsbdDKS17cZBz_nNtqWgyQU7JhHWe2V-1beM_8U3Q4_tHXFlVaE6UJeD4ZvsWRwkNtBSLnE_J3JKdDJDdOq76_gOya90iWtW0=s0-d)
and sesquilinearity of the scalar product:
![\langle e^{ikx}|e^{ilx}\rangle = \langle V(b) e^{ikx}|V(b)e^{ilx}\rangle = e^{ib(l-k)}\langle e^{ikx}|e^{ilx}\rangle](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vwSz_caCzw4UYcO6a0GPhbGeuC1FaX030dEmwTX43lsg713jD80_QuWyjdShw4Mj2XOGc0UUw2ceUsot_vFC85gCe_3HWnjjo6C3oMrokUjyqs_EpDquE8Q2QJ5_-3Y9T32iA-Tddlt3QVqHqjHNRKtgTEeSuzIPuekIQ-qDFihjKMT8O9K7huNA49vlxL9NkcYdnTAv5r1CUlWXJcAy2iHawX96rYEWrErpErv5eCxB26PDytoZHDoxT3GzwSyLcLPfavyeZj2FSyzVTsmvUe6l7FRFqJ89UUC2bZc9C1bzuNmSbkVZZtfF6kuDKrb3EK3w0HsRDZKct7tZqN0xLDHOisZz0Q=s0-d)
. This has to hold for all
![b](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sZTfEKoTD2r7BfHjWph0MmgXC2vBwR_BgUyh0gBbRpOQ1oDWF8adUEm7fcS9wdhZay3ARFyV8qwUdNHhc4wllrbFui8jqr4ekeI5SJYVhJmxQF0Q8=s0-d)
and thus if
![k\ne l](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tJVRn2vQHSKPEHJkx2uUkUYI2UakizZzFvc_A3KccexAswHUQnXKyhXEQXNmME_x528x9WADQnPtDgjeZQwsVoYtseXrPYkN4O3zccd339c6ksVTDTr7C4Rio=s0-d)
it follows that the scalar product vanishes.
Thus we have found our (polymer) Hilbert space: It is the space of (square summable) linear combinatios of plane waves with a scalar product such that the
![e^{ikx}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_usXIIhFgb715mDrNJbKFTclxc5e-uf5xLyVBuCOn2qLtFda17YXydKXYcqjs8imDk-U8EhaKpK6IK2GS-9FW9_f0RqvUTc8htOf21d6pQz0VqUgx5VJCgLNBvrtXyiGys=s0-d)
are an orthonormal basis.
Now, what about
![x](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sDlGnqF3SBi11RxxWz7OVnE_sbsUnN54AfKdenNHFU7Mp3mmGHN3DhMGKn5wHuO_xm8Q4KCM-9bQKYLCKhUsUPMqQMDMY1R0sT9PnsRCDOtsAkTA=s0-d)
and
![p](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uJIEc563cadzGfEL9f_soRts90lqstFT93-E6uWAUNpj2ksrv9_mAXdu0M3Ck2S_gMJFZsA_RWN5w6tHiXf7bdODpGuL6-b-mTexo5m33l9_3tPQ=s0-d)
? It is easy to see that
![p](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uJIEc563cadzGfEL9f_soRts90lqstFT93-E6uWAUNpj2ksrv9_mAXdu0M3Ck2S_gMJFZsA_RWN5w6tHiXf7bdODpGuL6-b-mTexo5m33l9_3tPQ=s0-d)
when defined by a derivative as above acts in the usual way, that is on a basis element
![pe^{ikx}=ke^{ikx}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tY_pRB38j2P3T2dDv14hm3dEVk1ovcZvw0zbU01P7i12AVcikGDBSycRxy6ATLu4WMNkTomPOUt4Vy2YX6Hu2hOBamRIJKT90Q9EuV8nSPuCtfeJ8fAI7BLv_kN0Soo-yvL13-bLgdWQRKVOYTrDYG=s0-d)
which is unbounded as
![k](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tO3YmQ5beGB4a6MlNDokgiKRwDmw3phzlkWxOQowNVA6CkUwtPzO7F-AVddC_FbZ3pIt6iDDF_oseFs9lkc0L9M9yLLmyXSUdeMQRnmne6IO1FLMk=s0-d)
can be arbitrarily large. The price for having plane waves as normalisable wave functions is, however, that
![x](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sDlGnqF3SBi11RxxWz7OVnE_sbsUnN54AfKdenNHFU7Mp3mmGHN3DhMGKn5wHuO_xm8Q4KCM-9bQKYLCKhUsUPMqQMDMY1R0sT9PnsRCDOtsAkTA=s0-d)
is not defined: It would be
![xe^{ikx} = \lim_{\epsilon\to 0}\frac{e^{i(k+\epsilon}x}-e^{ikx}}{\epsilon}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tbXhpsDLpwk8U_WJBstFDL5445prLAxteDv9j9Eu3Rw-aEQn2STWDfgD7NlBYwFYJcAGqHtleG7-s6MvaPGk6VDDD_QuX3dJMn4RZmWJCKuaJELzoDOcRZE15uNQo9QxMyDwjrmjyZ0pBciG081l42QvNar_IkVEzAUS10KMabxsJhILRrHFccVb7rJa53QIeeO5CFeySuD0dyNDg0QjjaK50lY7A-t9IXNodSGKqC0_tCaawvQAItSHnp4oPbPOvFL_uv_95kQg=s0-d)
. But for
![\epsilon\ne 0](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tlIWwE4kiC8goiBienp5oUudpW0BFGhIoU902lIztHUrfUAYXDs50vEBlyFLPml1yQdSFWoOtcSYpwYxenRssPdSk5Xe2FrUe0POu_1Ic-IUbXDAev9vwggbLdPNIecvAYB2vG=s0-d)
the two exponentials in the denominator are always orthogonal and thus not "close" as measured by the norm. The denominator always has norm 2 and thus the limit is divergent. Another way to see this is to notice that
![x](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sDlGnqF3SBi11RxxWz7OVnE_sbsUnN54AfKdenNHFU7Mp3mmGHN3DhMGKn5wHuO_xm8Q4KCM-9bQKYLCKhUsUPMqQMDMY1R0sT9PnsRCDOtsAkTA=s0-d)
would of course act as multiplication by the coordinate
![x](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sDlGnqF3SBi11RxxWz7OVnE_sbsUnN54AfKdenNHFU7Mp3mmGHN3DhMGKn5wHuO_xm8Q4KCM-9bQKYLCKhUsUPMqQMDMY1R0sT9PnsRCDOtsAkTA=s0-d)
, but
![x](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sDlGnqF3SBi11RxxWz7OVnE_sbsUnN54AfKdenNHFU7Mp3mmGHN3DhMGKn5wHuO_xm8Q4KCM-9bQKYLCKhUsUPMqQMDMY1R0sT9PnsRCDOtsAkTA=s0-d)
times a plane wave is no longer a linear combination of plane waves.
To make contact with loop cosmology one just has to rename the variables: What I called
![p](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uJIEc563cadzGfEL9f_soRts90lqstFT93-E6uWAUNpj2ksrv9_mAXdu0M3Ck2S_gMJFZsA_RWN5w6tHiXf7bdODpGuL6-b-mTexo5m33l9_3tPQ=s0-d)
for a simplicity of presentaion is the volume element
![v](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_umbS0LWdcapwFJX67CLzx_zFOdYgIqQER1P90l9WulRpP4HPGBYskTor6Jecv6oLVWbKirdrXtYcKSbfAxYJBrIVqEBos4v6oxCFzaOYeqBBeun34=s0-d)
in loop cosmology while the role of
![x](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sDlGnqF3SBi11RxxWz7OVnE_sbsUnN54AfKdenNHFU7Mp3mmGHN3DhMGKn5wHuO_xm8Q4KCM-9bQKYLCKhUsUPMqQMDMY1R0sT9PnsRCDOtsAkTA=s0-d)
is played be the conjugate momentum
![\beta](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_t1S3pfEhIwfflIvfrW5ClbDlLpI2zsdhTfmJ277kIcuE69WCDeJr93RD9UNvIndwhmLdBmuE2CxDsVlGH4t9BwCDTERq3AIyhU734RR35mGifzO3JJE3qW-Ts=s0-d)
.
If you want you can find my notes for the blackboard talk at HU here (
pdf or
djvu
8 comments:
It's interesting to note that these issues were discussed in a series of papers in the 50's and 60's by Schwinger.
The paper that is most relevant to the discussion at hand is
``Unitary transformations and the action principle'', http://www.pnas.org/content/46/6/883.full.pdf
Robert,
I do not understand this part:
"Thus we declare the constant wave function psi(x)=1 to be an element of the Hilbert space and we can assume that it is normalised"
Does this not already contain a problem for x?
What do you mean by "a problem for x"? Of course, the constant is not normalised with respect to the usual L^2 measure, but we specify the appropriate polymer measure below.
It seems that phi=1 is an eigenstate of p (eigenvalue 0) but also an eigenstate of x. But I am perhaps confused about that...
I should be more clear: If I use your formula for k=0 I get
x*1 = [(1+iex) - 1]/e
with e being your epsilon.
and thus x*1 = i*x which implies x=0.
Nice post Robert. Last time I came to say hello I was confused in spelling polymer versus pohlmeyer lol!
Post a Comment