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It's complicated...

There are two equivalent ways to represent the universe:

1) The average value of the Higgs field is zero and there is a HUGE amount of Higgs bosons everywhere, but all the calculations are horribly^1000000 difficult.

2) The average value of the Higgs field is a constant that is not zero and there are very few Higgs bosons here and there, and the calculations are easy [1].

Obviously physicist prefer the second description, in spite both are equivalent.

There are some technical problems if you imagine that there is a constant everywhere in the universe, and has exactly the same value [2]. So the solution is that it has the "constant" has an average value and allow local variations. The local variations are the Higgs bosons, because the field is quantized.

In a universe where the average value of the Higgs boson is not zero, but there are no Higgs bosons, you get the same technical problems that were solved with the idea of Higgs.

[1] It's easy if you have a PhD in physics, a few years of specialization. I can't do them, but I know people that can.

[2] There are other constants anyway, but they are different... I have no better way to explain it :(.




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