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L =½        + ½        ½(2 v  )  2

L =½        + ½        ½(2 v  )  2  v    2 +  2 ¼  2 +  2   + ¼ v 4. ½         ½(2 v  )  2. Explicitly expressed in real quantities  and v this is now an ordinary mass term !.  “appears” as a scalar (spin=0)

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L =½        + ½        ½(2 v  )  2

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  1. L=½ + ½ ½(2v)2 v2+2¼2+2  + ¼v4 ½  ½(2v)2 Explicitly expressed in real quantities  and v this is now an ordinary mass term!  “appears” as a scalar (spin=0) particle with a mass ½  “appears” as a massless scalar There is NO mass term!

  2. Of course we want even this Lagrangian to be invariant to LOCAL GAUGE TRANSFORMATIONS D=+igG Let’s not worry about the higher order symmetries…yet… free field for the gauge particle introduced Recall: F=G-G  *  12 + i22 again we define:  1 + i2

  3. Exactly the same potentialU as before! so, also as before: Note: v =0 with

  4. L= [+ v22] + [  ] + [ FF+ GG] -gvG 1 2 g2v2 2 1 2 -1 4 +{ g2 2 gG[-] + [2+2v+2]GG  2 + [2v3+v4+2v2 1 2 - (4+43v+62v2+4v3+4v2 + 222+2v22+v4 + 4 ) ]} L= [+ v22] + [  ] + [ FF+ GG] -gvG 1 2 g2v2 2 1 2 -1 4 g2 2 +{ gG[-] + [2+2v+2]GG  4  2 1 2 - [4v(3+2) + [ v4] + [4+222+ 4] and many interactions between  and  which includes a numerical constant v4 4

  5. The constants , v give the coupling strengths of each                  

  6. which we can interpret as: massless scalar  scalar field  with free Gauge field with mass=gv L = + + a whole bunch of 3-4 legged vertex couplings - gvG + But no MASSLESS scalar particle has ever been observed  is a ~massless spin-½ particle is a massless spin-1particle spinless,have plenty of mass! plus - gvG seems to describe G  • Is this an interaction? • A confused mass term? • G not independent? ( some QM oscillation between mixed states?) Higgs suggested:have not correctly identified the PHYSICALLY OBSERVABLE fundamental particles!

  7. Note: • Remember L isU(1)invariant • rotationally invariant in , (1, 2) space – • i.e. it can be equivalently expressed • under any gauge transformation in the complex plane or /=(cos + isin )(1 + i2) =(1cos-2sin ) + i(1sin+ 2cos) With no loss of generality we are free to pick the gaugea , for example, picking: /2  0 and/ becomes real!

  8. ring of possible ground states 2  1 equivalent to rotating the system by angle -  (x) (x) = 0

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