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_v_UG7ofbEKntngnK-tFwQ_azwraAXEbdnUh3lwGVBawYbiQqzGDbfHj9r7Zx9SBbEA_q18kmDO2Ez6i2f_k045rk96Wr4vkhwxIe062VNsCn1xJA=s0-d)
and
![p](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vhuIPTC04MVACMIrAiBPCSM2QJQ2ndgGNlJFUdXnKDyV9AgrGhcRJtrRqzY-J-YySmgWTYKBf_lNix_X2iLWDnEqXs4x5sHtOyT5jFoiGAOk8-Eg=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_tXRcvHbAE8Sdy9u5wv-O-CHeoxLaZSpZ8wk6zlFWSy0IgmolNpkmPEy4RY1UHSpZ2zqCGa3IgiGEN5sc3gAjNOxH_A46sT4CxxArQqpFWMA7yMsbPvjnH02THoAxfG=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_tMaUf8YOgeo4eLuV4ryqipO58sO0yOoz_PiKE9X9CBfqu8LmNip_b0PiBSabEriiQPQ-amRMACdE6tX2LFli4DaPYdB6123gtq_vb4ARiEPSJK9DIK_G8AmmGoXVGyf5r_Jk8=s0-d)
and
![V(b)=\exp(ibp)](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sgdaLx0NJzit9ZrCRuRyonFsUoylKzU7tF84id2vnT_wV_NfrSeuTR0u3mCoTwTV0yCitKEZNK9on__PgOJoEitz-Yw2B7Yht_1hOqyt_88AAjLC_LMtkdhOwVyi-JY9VwPB4=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_tXKo2u4X-Heci6fEdYkw6iTZhRKHY80TSg3oYjgLgfyd6qUdnJ64jFSgic5Tt4k84xuf2GcYtu4JM-TDBWh2cUK7G6K8s1Y1XKYdafu4uxVR5gK0wnuZlFMcBmX0aS2whBl-9nkTN-3ol-HvQPrexj_g=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_us6BsbVNwt0m7KnNgPPrWkUI9JgJ0sPYFxoaxIniAiI94xpbXi9QRO_Dvvr1zQmbE_uZfqbaLVvvhXWXR1yDyZ7xgE9Lo-Y09yCK9qM08_y11fypuRSOYM5FhNWN104iPDPEV9AQ8sw9CpmGs=s0-d)
and similar for
![p](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vhuIPTC04MVACMIrAiBPCSM2QJQ2ndgGNlJFUdXnKDyV9AgrGhcRJtrRqzY-J-YySmgWTYKBf_lNix_X2iLWDnEqXs4x5sHtOyT5jFoiGAOk8-Eg=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_uYZGSx5fSLKIbwZwS_gyf52xpfh7QQJytBXZIp8QcbH06FxWGr8NcrrTNBfBHtYj92jNC8gVs0VW1HloJ3ZF23znQ3HAIhUQQeH2e7kM7HmKbJx3I=s0-d)
and
![V](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vmE5nHoPX29WipUXR5wlKWMsV4C3_nIy-EqkLt6rWBEZwYaJq5OzVZ_heXhHy75tZ8tsB0H2L5RO_NmQZg9Fo-KTVCSzBf4nArNo1wT-PqcyJK7yg=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_vCHmCfDDoqb3kFRKg3qLeaEsAGKhKLd_dUsPHyTwloQy5CC8gTNrnTdZN8CRHabpq3XCG2EEvMYSQE2IUqQqcGtqbzGnNjt3CZaQkVx8P0J9MnAOPmwnX-G64O_g=s0-d)
.
![U](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uYZGSx5fSLKIbwZwS_gyf52xpfh7QQJytBXZIp8QcbH06FxWGr8NcrrTNBfBHtYj92jNC8gVs0VW1HloJ3ZF23znQ3HAIhUQQeH2e7kM7HmKbJx3I=s0-d)
and
![V](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vmE5nHoPX29WipUXR5wlKWMsV4C3_nIy-EqkLt6rWBEZwYaJq5OzVZ_heXhHy75tZ8tsB0H2L5RO_NmQZg9Fo-KTVCSzBf4nArNo1wT-PqcyJK7yg=s0-d)
then act in the usual way,
![V(b)](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vi6yYX48KsGIjXXHya_8Qtl9sp9IWurp30GZoWPumyywJBvm0i462-fALZkHgxjMqy76j0h1pnUM6wf2O2WWvWVGq7CzCxGwH6KYWjEfGiH4Eni9_oNpc=s0-d)
by translation
![(V(b)\psi)(x)=\psi(x-b)](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sMBG334pk4cIa20ahNt_0Fw3TMXuQnZc2t5PKeEhICUBpj4BEDZmyAidPWDj-1d8rTAQw4dEDZ1Vp-qNSTGz84oCusMYAJ-fC8g4oVzEi6B3Jnw3-whtUagU0q_OBXEj-eE74vZEdwoh5z6C0l=s0-d)
and
![U(a)](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vwFkLj9aRHbiewzqzuxIWkbYGyOOPXnYsbDmCGQeqTpMsZzjGaoSvJhE45t_DTUNZpCeIFsmMmkctm062xVzOGlYUzW1FMfYGsRBo7aq31FU1u6iCQ4q0=s0-d)
by multiplication
![(U(a)\psi)(x)=e^{iax}\psi(x)](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vojmjriFJ7Od21A5YLY2ZIr1Y0axVLG8KAX1pR34On5pcPVz1FGul6RA-IA53RXmyL0dQkEZMVemJtQuhiphDZSj5K4R24MEEQJdZHS7xlICariZSZpnr-9fFmBm5LO_1Iao2GsITKBPAQzkqSc52BIMjVAzarQWWI=s0-d)
. Obviously, these fulfil the commutation relation. You can think of
![U](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uYZGSx5fSLKIbwZwS_gyf52xpfh7QQJytBXZIp8QcbH06FxWGr8NcrrTNBfBHtYj92jNC8gVs0VW1HloJ3ZF23znQ3HAIhUQQeH2e7kM7HmKbJx3I=s0-d)
and
![V](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vmE5nHoPX29WipUXR5wlKWMsV4C3_nIy-EqkLt6rWBEZwYaJq5OzVZ_heXhHy75tZ8tsB0H2L5RO_NmQZg9Fo-KTVCSzBf4nArNo1wT-PqcyJK7yg=s0-d)
as the group elements of the Heisenberg group while
![x](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_v_UG7ofbEKntngnK-tFwQ_azwraAXEbdnUh3lwGVBawYbiQqzGDbfHj9r7Zx9SBbEA_q18kmDO2Ez6i2f_k045rk96Wr4vkhwxIe062VNsCn1xJA=s0-d)
and
![p](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vhuIPTC04MVACMIrAiBPCSM2QJQ2ndgGNlJFUdXnKDyV9AgrGhcRJtrRqzY-J-YySmgWTYKBf_lNix_X2iLWDnEqXs4x5sHtOyT5jFoiGAOk8-Eg=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_tBhRgcu6L-RDl7QALDCxjJewRnHXKzjS8qXadl-z-ZB5tSb3BsvrlmgfLyTCWJ_bWSG4XwpDaFT4py5qqjIeuRAqByQUOQZqMIlmKYe-9WC7TkoGBWNPgxPiYvd5_IQ643COwPK0U4jGfOTUzsUg=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_sz-V23335WYGPXEVxXGECCDU8McABsin4LMK57W-bEKc3ZGTHEhe_u5h3s3BMiA-b8zNz84dVIcPqMS3-M0TWRrz4B0u0iBPWwNGA1Sjbb-mXJJJF5VsxhkzmSpqERP5eFLe6VhAZyYhmgzOej=s0-d)
.
Acting now with
![U(a)](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vwFkLj9aRHbiewzqzuxIWkbYGyOOPXnYsbDmCGQeqTpMsZzjGaoSvJhE45t_DTUNZpCeIFsmMmkctm062xVzOGlYUzW1FMfYGsRBo7aq31FU1u6iCQ4q0=s0-d)
, we find that linear combinations of plane waves
![e^{ikx}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_t7AYA_K0alDd1-iUl_CPIFtfYeYXwqVtq8bM4CIuBydE2Q49RT0S4ALUiGElcqqCuL3cYb03ucC7-iXGGQNNQwmo-D5Ysk9mdPCa-F536PcG7b3QLfGo5ljHz4TFhxowM=s0-d)
are then as well in the Hilbert space. By unitarity of
![U(a)](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vwFkLj9aRHbiewzqzuxIWkbYGyOOPXnYsbDmCGQeqTpMsZzjGaoSvJhE45t_DTUNZpCeIFsmMmkctm062xVzOGlYUzW1FMfYGsRBo7aq31FU1u6iCQ4q0=s0-d)
, it follows that
![\langle e^{ikx}| e^{ikx}\rangle =1](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_taSJA7Hx8eaqFqMuLb_xY3UdslJ7kTFJ7QEcabA488dAW6s9i6wUOxa5ziBYdY4XcDavZSWa4yPFfjAj3JeKrn8dtGsM2g2sGLEXsSJj4uLMKEdl6bFHli-VstTqB6ZpMGAeBTk7zPraqFD8Hz9rPoG4KGu7qw4vHGLoGArZTIMyI2vDM8PRU=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_sCLqVfcQvbr7Krs4_0A23inkqH_VHt92cZzxpdmF-gjcL4OpF_wuH8ZuxEyUuqmbIfg2KtYjaBWX55XJ8zz7GrxFBtkCp5Enyi-a7JvUyrs5ks3T_4PHUG5fnLWqWgyTxVlzA6yz8ViZc_uaUR4iEA5n5qNTu4ZmdGKfgMQv5zvQwNWQ=s0-d)
. This is found using the unitarity of
![V](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vmE5nHoPX29WipUXR5wlKWMsV4C3_nIy-EqkLt6rWBEZwYaJq5OzVZ_heXhHy75tZ8tsB0H2L5RO_NmQZg9Fo-KTVCSzBf4nArNo1wT-PqcyJK7yg=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_s0Z251h72-nK7DKmdePqavy9N2S27YpOHjZLO0dmbt2VOuLVF0k18ln22ts6ztl_Gz_XFUIpjMF7bkF5PNXPmaNDVNEG99yXEv-r2iB8iO7x5YUQBKfRUrYCdtrFmuFcraYuBcSTMlYyFfENGnmmeuatrqk-OzwkN8dOH1F5-EAdOPo6VaP1HizY-msUcdbTjgCB4vdgpJH1fjEOzjM6pjLIvOlYJ6LGMcvCyEZvRlrSm12yy-4hwEppopO9hek_ZOWEA81_wpELXgKwIrYIo6qzyU3nL6lbeQyP3wjPQmzgNGVehxHRDm6_FXwtkH9ax0ZJ8kmgVL6yRmJduVwkDbtQEi-fkf=s0-d)
. This has to hold for all
![b](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vWdM3nG9-pLp3uUlBz-YJNlQxXogiXcgou_s5nx8nfPICY-2XjcZ-RTRTgE0iprvNjSYYZD-OFp4Ut7kD7LBhrB9OGFl3tIxdrYoJId6Py7D9HTyQ=s0-d)
and thus if
![k\ne l](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vRWMd2h0u4aGXEr0Qw1eWD7a2ov-RczU2iX3rmB2luEFnqJ-54dg5yIoIlScJxe9XOJe_2pPN7L5X_jAIQ1h5REdkVotwyDmeFCLNRKg7SUx0XQ8gnptwZa0Q=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_t7AYA_K0alDd1-iUl_CPIFtfYeYXwqVtq8bM4CIuBydE2Q49RT0S4ALUiGElcqqCuL3cYb03ucC7-iXGGQNNQwmo-D5Ysk9mdPCa-F536PcG7b3QLfGo5ljHz4TFhxowM=s0-d)
are an orthonormal basis.
Now, what about
![x](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_v_UG7ofbEKntngnK-tFwQ_azwraAXEbdnUh3lwGVBawYbiQqzGDbfHj9r7Zx9SBbEA_q18kmDO2Ez6i2f_k045rk96Wr4vkhwxIe062VNsCn1xJA=s0-d)
and
![p](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vhuIPTC04MVACMIrAiBPCSM2QJQ2ndgGNlJFUdXnKDyV9AgrGhcRJtrRqzY-J-YySmgWTYKBf_lNix_X2iLWDnEqXs4x5sHtOyT5jFoiGAOk8-Eg=s0-d)
? It is easy to see that
![p](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vhuIPTC04MVACMIrAiBPCSM2QJQ2ndgGNlJFUdXnKDyV9AgrGhcRJtrRqzY-J-YySmgWTYKBf_lNix_X2iLWDnEqXs4x5sHtOyT5jFoiGAOk8-Eg=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_vOojzo_tDk4zx0JW2Exj3BvRFSoHk5KLK0NCQMUalcEPRMwNyOEgpGw9eEbfhKjdryaRtqFiRLbn1t1e4GFw-gwfK_jE533HFIBs0lbTCVObru1CikYXQpZWl-EqdPPxUtE35juf67AeoZBv6tQxIe=s0-d)
which is unbounded as
![k](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_syST3HwnBAP7Ftuvsp_yr_WnAMf2_b5H0zMd-p8Ad1ROtOaMjdCIjbQGzyxzywkD_BHGOQe8kl--HPSzyfWBhOybGrTk_sJGcbZ-31RFEyOxeFVYM=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_v_UG7ofbEKntngnK-tFwQ_azwraAXEbdnUh3lwGVBawYbiQqzGDbfHj9r7Zx9SBbEA_q18kmDO2Ez6i2f_k045rk96Wr4vkhwxIe062VNsCn1xJA=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_ux1Anr3fW1r8zK7qwva1sQtALW26_XXVFYqbtXkQi5V4ikJhSrxKmGs_AQemSIWQeLvp-fSJDiB3BMu_UwENFW1NoJjPHXEYUy-NxSR5eJR-VSeA2-eQ2ehsDRCSD05_pyZ7kMJ0Fs2_FqCBIcCO0jNZHVug86YqpRwrOcogLscSGAByMfwBnwZ12JBvbThXj2D34e6e3uyWkDuvO3GUWFUQ7RgTXWHaRmLCSE2zHZ9A9rm_XMPjW1RY67u-8vwvDBIQ7NbwMglQ=s0-d)
. But for
![\epsilon\ne 0](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vgsJorA0OwaQD9rgFrvL4F1Lr8K--KIhp9WgpUweM6NsyfW_UM-dvxE2Q0v-u9ZepOYquUAcLNxsqNMjJfSzRIKGYurQBbNU57RWFVBJg2kalyX9KILiMc98Wt-WXAQh74buiZ=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_v_UG7ofbEKntngnK-tFwQ_azwraAXEbdnUh3lwGVBawYbiQqzGDbfHj9r7Zx9SBbEA_q18kmDO2Ez6i2f_k045rk96Wr4vkhwxIe062VNsCn1xJA=s0-d)
would of course act as multiplication by the coordinate
![x](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_v_UG7ofbEKntngnK-tFwQ_azwraAXEbdnUh3lwGVBawYbiQqzGDbfHj9r7Zx9SBbEA_q18kmDO2Ez6i2f_k045rk96Wr4vkhwxIe062VNsCn1xJA=s0-d)
, but
![x](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_v_UG7ofbEKntngnK-tFwQ_azwraAXEbdnUh3lwGVBawYbiQqzGDbfHj9r7Zx9SBbEA_q18kmDO2Ez6i2f_k045rk96Wr4vkhwxIe062VNsCn1xJA=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_vhuIPTC04MVACMIrAiBPCSM2QJQ2ndgGNlJFUdXnKDyV9AgrGhcRJtrRqzY-J-YySmgWTYKBf_lNix_X2iLWDnEqXs4x5sHtOyT5jFoiGAOk8-Eg=s0-d)
for a simplicity of presentaion is the volume element
![v](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_taoh7ZxE3V2ZkDbta_UN8gWPfzn2ZAch78kkYiPnGDlRgKOko8AC1o73je98bsTZs1zAOwhnD523JKHuqjHHXbyUyL00Cs5MFjd0EiR3cP6QCmeog=s0-d)
in loop cosmology while the role of
![x](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_v_UG7ofbEKntngnK-tFwQ_azwraAXEbdnUh3lwGVBawYbiQqzGDbfHj9r7Zx9SBbEA_q18kmDO2Ez6i2f_k045rk96Wr4vkhwxIe062VNsCn1xJA=s0-d)
is played be the conjugate momentum
![\beta](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tn4R3-cVgOZUK7cKstjn0eNV2sH5-zdWF3Q7MZ1TUo8NdqBBE6BFUanLylPAau4xyI8amAC-4NNJH7h3gYEMJthT6CI7qFMN2L40ts72s4pm-hwxW5o5wZDco=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!
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