There are reasons we are not interested in a vacuum cleaner at the moment. I'm sure the same applies for other people. Is there an alternative that can get rid of dirt as well as pick up bigger particles?
Use a carpet sweeper.
Bissell sells several models of mechanical manual sweepers -- an invention that's been around in some form since the 1880s. They work, but only to an extent. For deep cleaning you'll have to use a vacuum or roll up your carpet and take it in to the pros.
In the "olden days" before there were vacuum cleaners you either swept (which still works to an extent -- tough bristles, short quick strokes are key) or took the carpet up and beat the dust and dirt out with a carpet beater (which also works, as long as your carpet isn't fitted).
Alternatively, there are machines that spray water and shampoo into the carpet and suck up the dirty water -- but these probably come under the heading of a "vacuum cleaner," and so aren't appropriate in your case. Also, they're not very useful for picking up bigger particles.
Beyond those options -- and keeping your carpet clean in the first place (Martha Stewart suggests lots of doormats) -- I don't think that there's another effective way of getting your carpet clean.
The purpose of carpet is two-fold. First, to make a nice, cushy surface for your delicate feet to tread upon. Second, as an insulating barrier from the extreme expressions of Maw Nature. The side-effect of carpet is a trap for disease, dead skin and all, plus an emitter of chemical emissions – a byproduct of materials used in the construction. The only logical resolution – to the chagrin of carpet manufacturers, is to rip it all out, insulate under the flooring and buy one of those wood floor sweeper-dusters as seen on TV. Hold out for the “wait, there is more” advertisements, best bang for the buck.
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As zero is a number, “you can’t” equally applies to “you cannot clean carpet without a vacuum.”
If you’ve never removed (ripped up and disposed of) carpet and/or padding in a regularly vacuumed environment, you are not qualified to answer, and especially not qualified to down-vote!! If you have removed carpet then you know, without a doubt, a vacuum of some type is essential to the maintaining a clean and healthy carpeted environment.
In the below diagram, please note the accumulation of dirt between the layers of the carpet, the padding and the flooring. These particles of dust (http://en.wikipedia.org/wiki/Dust) dead skin (http://www.thenakedscientists.com/HTML/questions/question/1643/) and itsy bitsy creatures such as dust mites (http://en.wikipedia.org/wiki/House_dust_mite) are just some of the components of the yucky stuff labeled “filth and dirt” in the cheap artwork attached to this post.
The OP’s question is “There are reasons we are not interested in a vacuum cleaner at the moment. I'm sure the same applies for other people. Is there an alternative that can get rid of dirt as well as pick up bigger particles?”
I propose the following answer : “no.”
Lint rollers work on clothes and hard surfaces, and certain products also work on carpets. Consider, for instance, something like this large sticky-tape roller, called the Roll-O-Vac.
For a thorough carpet clean, you'll have to use a plug-in machine or call in the pros.
Buy carpet squares. Then take carpet squares outside and beat them. Then throw them in the wash machine.
Don't know how doable for you this is (but I think I am the only person to actually answer the question fully). I have seen people get out stains in carpet squares through washing so why not the whole floor...